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Life of π: Continued Fractions and Infinite Series
1. Outlines
Life of π: Continued Fractions and Infinite Series
Daniel J. Hermes
February 29, 2012
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
2. Outlines
1
π =3+
9
6+
25
6+
49
6+
.
6 + ..
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
3. Part I: Introductory Facts
Outlines
Part II: What We Came Here For: π
Outline of Part I
Introduction
The Wrong Way
Smart Men
Conway and Pell
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
4. Part I: Introductory Facts
Outlines
Part II: What We Came Here For: π
Outline of Part I
Introduction
The Wrong Way
Smart Men
Conway and Pell
Continued Fractions
Define It
What Does it Mean to Converge?
General
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
5. Part I: Introductory Facts
Outlines
Part II: What We Came Here For: π
Outline of Part I
Introduction
The Wrong Way
Smart Men
Conway and Pell
Continued Fractions
Define It
What Does it Mean to Converge?
General
Working with the Convergents
Difference
Telescoping Partials
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
6. Part I: Introductory Facts
Outlines
Part II: What We Came Here For: π
Outline of Part I
Introduction
The Wrong Way
Smart Men
Conway and Pell
Continued Fractions
Define It
What Does it Mean to Converge?
General
Working with the Convergents
Difference
Telescoping Partials
Lemmata: Sums for π
Review
Smith Sum
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
7. Part I: Introductory Facts
Outlines
Part II: What We Came Here For: π
Outline of Part II
Analysis and Arithmetic
A Nifty Identity
Denominator Series (kn )
Convergence
Plug and Play
Plug and Play
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
8. Part I: Introductory Facts
Outlines
Part II: What We Came Here For: π
Outline of Part II
Analysis and Arithmetic
A Nifty Identity
Denominator Series (kn )
Convergence
Plug and Play
Plug and Play
Parting Words
4 12
π and π 2
Thank You
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
9. Intro
CF
Convergents
Lemmata: Sums for π
Part I
Introductory Facts
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
10. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Introduction
The Wrong Way
Smart Men
Conway and Pell
Continued Fractions
Define It
What Does it Mean to Converge?
General
Working with the Convergents
Difference
Telescoping Partials
Lemmata: Sums for π
Review
Smith Sum
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
11. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
The Wrong Way
In bill #246 of the 1897 sitting of the Indiana General Assembly, π
was rational:
http://en.wikipedia.org/wiki/Indiana Pi Bill
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
12. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
The Wrong Way
In bill #246 of the 1897 sitting of the Indiana General Assembly, π
was rational:
...Furthermore, it has revealed the ratio of the chord and
arc of ninety degrees, which is as seven to eight, and also
the ratio of the diagonal and one side of a square which
is as ten to seven, disclosing the fourth important fact,
that the ratio of the diameter and circumference is as
five-fourths to four...
http://en.wikipedia.org/wiki/Indiana Pi Bill
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
13. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
The Wrong Way
In bill #246 of the 1897 sitting of the Indiana General Assembly, π
was rational:
...Furthermore, it has revealed the ratio of the chord and
arc of ninety degrees, which is as seven to eight, and also
the ratio of the diagonal and one side of a square which
is as ten to seven, disclosing the fourth important fact,
that the ratio of the diameter and circumference is as
five-fourths to four...
4
π= 5/4
= 3.2
http://en.wikipedia.org/wiki/Indiana Pi Bill
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
14. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Smart Men: Archimedes
In the third century BCE, Archimedes proved the sharp
inequalities
223/71 < π < 22/7
by means of regular 96-gons
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
15. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Smart Men: Archimedes
In the third century BCE, Archimedes proved the sharp
inequalities
223/71 < π < 22/7
by means of regular 96-gons
22 1
=3+
7 7
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
16. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Smart Men: The Bible
In 1 Kings 7:23 the word translated ’measuring line’
appears in the Hebrew text ... The ratio of the numerical
values of these Hebrew spellings is 111/106. If the putative
value of 3 is multiplied by this ratio, one obtains 333/106
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
17. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Smart Men: The Bible
In 1 Kings 7:23 the word translated ’measuring line’
appears in the Hebrew text ... The ratio of the numerical
values of these Hebrew spellings is 111/106. If the putative
value of 3 is multiplied by this ratio, one obtains 333/106
333 15
=3+
106 106
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
18. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Smart Men: The Bible
In 1 Kings 7:23 the word translated ’measuring line’
appears in the Hebrew text ... The ratio of the numerical
values of these Hebrew spellings is 111/106. If the putative
value of 3 is multiplied by this ratio, one obtains 333/106
333 15
=3+
106 106
1
= 3 + 106
/15
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
19. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Smart Men: The Bible
In 1 Kings 7:23 the word translated ’measuring line’
appears in the Hebrew text ... The ratio of the numerical
values of these Hebrew spellings is 111/106. If the putative
value of 3 is multiplied by this ratio, one obtains 333/106
333 15
=3+
106 106
1
= 3 + 106
/15
1
=3+
1
7+
15
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
20. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Smart Men: Zu Chongzhi
The 5th century Chinese mathematician and astronomer
Zu Chongzhi ... gave two other approximations ... 22/7
and 355/113
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
21. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Smart Men: Zu Chongzhi
The 5th century Chinese mathematician and astronomer
Zu Chongzhi ... gave two other approximations ... 22/7
and 355/113
355 16
=3+
113 113
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
22. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Smart Men: Zu Chongzhi
The 5th century Chinese mathematician and astronomer
Zu Chongzhi ... gave two other approximations ... 22/7
and 355/113
355 16
=3+
113 113
1 1
= 3 + 113 = 3 +
/16 7 + 1/16
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
23. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Smart Men: Zu Chongzhi
The 5th century Chinese mathematician and astronomer
Zu Chongzhi ... gave two other approximations ... 22/7
and 355/113
355 16
=3+
113 113
1 1
= 3 + 113 = 3 +
/16 7 + 1/16
1
=3+
1
7+
15 + 1/1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
24. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Smart Men: James Gregory
In 1672, James Gregory wrote about a formula for calculating the
angle given the tangent x:
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
25. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Smart Men: James Gregory
In 1672, James Gregory wrote about a formula for calculating the
angle given the tangent x:
x3 x5 x7
arctan x = x − + − + ···
3 5 7
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
26. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Smart Men: James Gregory
Why is this relevant? We can use it to approximate π!
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
27. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Smart Men: James Gregory
Why is this relevant? We can use it to approximate π!
∞
π 1 1 1 (−1)n
= 1 − + − + ··· =
4 3 5 7 2n + 1
n=0
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
28. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Smart Men: James Gregory
Why is this relevant? We can use it to approximate π!
∞
π 1 1 1 (−1)n
= 1 − + − + ··· =
4 3 5 7 2n + 1
n=0
1
Proof: Since d
dx arctan x = = 1 − x 2 + x 4 − x 6 + · · · and
1 + x2
arctan 0 = 0, integrating, we find the Taylor series expansion for
arctan is
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
29. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Smart Men: James Gregory
Why is this relevant? We can use it to approximate π!
∞
π 1 1 1 (−1)n
= 1 − + − + ··· =
4 3 5 7 2n + 1
n=0
1
Proof: Since d
dx arctan x = = 1 − x 2 + x 4 − x 6 + · · · and
1 + x2
arctan 0 = 0, integrating, we find the Taylor series expansion for
arctan is
x3 x5 x7
arctan x = x − + − + ···
3 5 7
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
30. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Smart Men: James Gregory
Why is this relevant? We can use it to approximate π!
∞
π 1 1 1 (−1)n
= 1 − + − + ··· =
4 3 5 7 2n + 1
n=0
1
Proof: Since d
dx arctan x = = 1 − x 2 + x 4 − x 6 + · · · and
1 + x2
arctan 0 = 0, integrating, we find the Taylor series expansion for
arctan is
x3 x5 x7
arctan x = x − + − + ···
3 5 7
∞
π (−1)n
Hence = arctan(1) = .
4 2n + 1
n=0
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
31. Intro
The Wrong Way
CF
Smart Men
Convergents
Conway and Pell
Lemmata: Sums for π
Conway and Pell
In the actual talk, I displayed “Hey Dan, go to the chalkboard
please.” and just talked to them. For some reference see:
blog.bossylobster.com/2011/07/continued-fractions-for-greater-good.html
blog.bossylobster.com/2011/07/continued-fraction-expansions-of.html
blog.bossylobster.com/2011/08/conways-topograph-part-1.html
blog.bossylobster.com/2011/08/conways-topograph-part-2.html
blog.bossylobster.com/2011/08/conways-topograph-part-3.html
blog.bossylobster.com/2011/08/finding-fibonacci-golden-nuggets.html
blog.bossylobster.com/2011/08/finding-fibonacci-golden-nuggets-part-2.html
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
32. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
Introduction
The Wrong Way
Smart Men
Conway and Pell
Continued Fractions
Define It
What Does it Mean to Converge?
General
Working with the Convergents
Difference
Telescoping Partials
Lemmata: Sums for π
Review
Smith Sum
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
33. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
Define It
Define a standard continued fraction with a series of nonnegative
integers {an }∞ :
n=0
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
34. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
Define It
Define a standard continued fraction with a series of nonnegative
integers {an }∞ :
n=0
1
a0 +
1
a1 +
1
a2 +
.
a3 + . .
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
35. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
Define It
How will we know if our fractions converge if we don’t have a
concept of a “convergent sum”?
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
36. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
Define It
How will we know if our fractions converge if we don’t have a
concept of a “convergent sum”? For a standard continued fraction
corresponding to {an }, let the partial convergent “stopping” at an
be the fraction hn/kn .
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
37. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
Define It
How will we know if our fractions converge if we don’t have a
concept of a “convergent sum”? For a standard continued fraction
corresponding to {an }, let the partial convergent “stopping” at an
be the fraction hn/kn . As we’ll show, the series {hn } and {kn } are
very related and can be used to determine convergence.
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
38. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
Define It
How will we know if our fractions converge if we don’t have a
concept of a “convergent sum”? For a standard continued fraction
corresponding to {an }, let the partial convergent “stopping” at an
be the fraction hn/kn . As we’ll show, the series {hn } and {kn } are
very related and can be used to determine convergence.
Notice the zeroth partial is a0 so h0 = a0 , k0 = 1.
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
39. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
Define It
How will we know if our fractions converge if we don’t have a
concept of a “convergent sum”? For a standard continued fraction
corresponding to {an }, let the partial convergent “stopping” at an
be the fraction hn/kn . As we’ll show, the series {hn } and {kn } are
very related and can be used to determine convergence.
Notice the zeroth partial is a0 so h0 = a0 , k0 = 1.
1 a1 (a0 )+1
The first is a0 + a1 = a1 (1)+0 , h1 = a1 a0 + 1, k1 = a1 .
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
40. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
Define It
How will we know if our fractions converge if we don’t have a
concept of a “convergent sum”? For a standard continued fraction
corresponding to {an }, let the partial convergent “stopping” at an
be the fraction hn/kn . As we’ll show, the series {hn } and {kn } are
very related and can be used to determine convergence.
Notice the zeroth partial is a0 so h0 = a0 , k0 = 1.
1 a1 (a0 )+1
The first is a0 + a1 = a1 (1)+0 , h1 = a1 a0 + 1, k1 = a1 .
1 a2 a2 (a0 a1 +1)+a0
The second is a0 + = a0 + a1 a2 +1 = a2 (a1 )+1 .
1
a1 +
a2
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
41. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
What Does it Mean to Converge?
Claim: hn and kn satisfy the same recurrence relation
hn = an hn−1 + hn−2
kn = an kn−1 + kn−2
along with initial conditions h−1 = 1, h−2 = 0 and
k−1 = 0, k−2 = 1.
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
42. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
What Does it Mean to Converge?
Proof: There is one key insight: when go from one partial to the
1
1
next by turning the final an into 1
. Using the recurrence
an + an+1
and the inductive assumption:
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
43. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
What Does it Mean to Converge?
Proof: There is one key insight: when go from one partial to the
1
1
next by turning the final an into 1
. Using the recurrence
an + an+1
and the inductive assumption:
hn an hn−1 + hn−2
=
kn an kn−1 + kn−2
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
44. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
What Does it Mean to Converge?
Proof: There is one key insight: when go from one partial to the
1
1
next by turning the final an into 1
. Using the recurrence
an + an+1
and the inductive assumption:
hn an hn−1 + hn−2
=
kn an kn−1 + kn−2
1
hn+1 an + an+1 hn−1 + hn−2
becomes =
kn+1 an + 1
kn−1 + kn−2
an+1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
45. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
What Does it Mean to Converge?
Proof: There is one key insight: when go from one partial to the
1
1
next by turning the final an into 1
. Using the recurrence
an + an+1
and the inductive assumption:
hn an hn−1 + hn−2
=
kn an kn−1 + kn−2
1
hn+1 an + an+1 hn−1 + hn−2
becomes =
kn+1 an + 1
kn−1 + kn−2
an+1
Why can we replace an straight-up?
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
46. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
What Does it Mean to Converge?
Proof: There is one key insight: when go from one partial to the
1
1
next by turning the final an into 1
. Using the recurrence
an + an+1
and the inductive assumption:
hn an hn−1 + hn−2
=
kn an kn−1 + kn−2
1
hn+1 an + an+1 hn−1 + hn−2
becomes =
kn+1 an + 1
kn−1 + kn−2
an+1
Why can we replace an straight-up? All the other terms depend
solely on a1 , . . . , an−1 .
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
47. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
Standard
1
hn+1 an + an+1 hn−1 + hn−2
and =
kn+1 an + 1
kn−1 + kn−2
an+1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
48. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
Standard
1
hn+1 an + an+1 hn−1 + hn−2
and =
kn+1 an + 1
kn−1 + kn−2
an+1
an+1 (an hn−1 + hn−2 ) + hn−1
=
an+1 (an kn−1 + kn−2 ) + kn−1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
49. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
Standard
1
hn+1 an + an+1 hn−1 + hn−2
and =
kn+1 an + 1
kn−1 + kn−2
an+1
an+1 (an hn−1 + hn−2 ) + hn−1
=
an+1 (an kn−1 + kn−2 ) + kn−1
an+1 hn + hn−1
= by the inductive assumption.
an+1 kn + kn−1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
50. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
Standard
1
hn+1 an + an+1 hn−1 + hn−2
and =
kn+1 an + 1
kn−1 + kn−2
an+1
an+1 (an hn−1 + hn−2 ) + hn−1
=
an+1 (an kn−1 + kn−2 ) + kn−1
an+1 hn + hn−1
= by the inductive assumption.
an+1 kn + kn−1
Clearly, the recurrence is determined by just two terms, so with a
simple check, h−1 = 1, h−2 = 0 and k−1 = 0, k−1 = 1
⇒ h0 = a0 , h1 = a0 a1 + 1 and k0 = 1, k1 = a1 as we’d wish.
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
51. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
General
If we look to our favorite Sloane number sequence (A001203), we
find that standard continued fractions don’t do anything helpful
with π:
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
52. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
General
If we look to our favorite Sloane number sequence (A001203), we
find that standard continued fractions don’t do anything helpful
with π:
1
π =3+
1
7+
1
15 +
1
1+
.
292 + . .
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
53. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
General
But...
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
54. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
General
But...
1
π =3+
9
6+
25
6+
49
6+
.
6 + ..
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
55. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
General
Let’s generalize our definition!
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
56. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
General
Too General?
Define a generalized continued fraction with two series of
nonnegative integers {an }∞ and {bn }∞ :
n=0 n=0
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
57. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
General
Too General?
Define a generalized continued fraction with two series of
nonnegative integers {an }∞ and {bn }∞ :
n=0 n=0
b1
a0 +
b2
a1 +
b3
a2 +
.
a3 + . .
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
58. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
General
Too General?
Again, let the partial convergent “stopping” at an be the fraction
hn/kn .
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
59. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
General
Too General?
Again, let the partial convergent “stopping” at an be the fraction
hn/kn .
Claim: hn and kn satisfy the same recurrence relation
hn = an hn−1 + bn hn−2
kn = an kn−1 + bn kn−2
along with initial conditions h−1 = 1, h−2 = 0 and
k−1 = 0, k−2 = 1.
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
60. Intro
Define It
CF
What Does it Mean to Converge?
Convergents
General
Lemmata: Sums for π
General
Too General?
Again, let the partial convergent “stopping” at an be the fraction
hn/kn .
Claim: hn and kn satisfy the same recurrence relation
hn = an hn−1 + bn hn−2
kn = an kn−1 + bn kn−2
along with initial conditions h−1 = 1, h−2 = 0 and
k−1 = 0, k−2 = 1.
We won’t prove it, but the key insight is (you guessed it) turning
an into an + bn+1 .
a
n+1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
61. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Introduction
The Wrong Way
Smart Men
Conway and Pell
Continued Fractions
Define It
What Does it Mean to Converge?
General
Working with the Convergents
Difference
Telescoping Partials
Lemmata: Sums for π
Review
Smith Sum
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
62. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Difference: Dn
In the actual talk, I displayed “Hey Dan, go back to the chalkboard
please.” and just talked to them. I have included it here since
there is no chalkboard.
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
63. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Difference: Dn
hn
We wish to find the value of the continued fraction: lim .
n→∞ kn
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
64. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Difference: Dn
hn
We wish to find the value of the continued fraction: lim .
n→∞ kn
n−1
h1 hn hr hr +1
Notice − = −
k1 kn kr kr +1
r =1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
65. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Difference: Dn
hn
We wish to find the value of the continued fraction: lim .
n→∞ kn
n−1
h1 hn hr hr +1
Notice − = − , with this it becomes clear it
k1 kn kr kr +1
r =1
is important to define the difference: Dn+1 = kn+1 hn − kn hn+1 .
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
66. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Difference: Dn
hn
We wish to find the value of the continued fraction: lim .
n→∞ kn
n−1
h1 hn hr hr +1
Notice − = − , with this it becomes clear it
k1 kn kr kr +1
r =1
is important to define the difference: Dn+1 = kn+1 hn − kn hn+1 .
We have D1 = k1 h0 − k0 h1 = a1 a0 − 1 · (a1 a0 + b1 ) = −b1 and
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
67. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Difference: Dn
hn
We wish to find the value of the continued fraction: lim .
n→∞ kn
n−1
h1 hn hr hr +1
Notice − = − , with this it becomes clear it
k1 kn kr kr +1
r =1
is important to define the difference: Dn+1 = kn+1 hn − kn hn+1 .
We have D1 = k1 h0 − k0 h1 = a1 a0 − 1 · (a1 a0 + b1 ) = −b1 and
Dn+1 = (an+1 kn + bn+1 kn−1 )hn − kn (an+1 hn + bn+1 hn−1 )
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
68. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Difference: Dn
hn
We wish to find the value of the continued fraction: lim .
n→∞ kn
n−1
h1 hn hr hr +1
Notice − = − , with this it becomes clear it
k1 kn kr kr +1
r =1
is important to define the difference: Dn+1 = kn+1 hn − kn hn+1 .
We have D1 = k1 h0 − k0 h1 = a1 a0 − 1 · (a1 a0 + b1 ) = −b1 and
Dn+1 = (an+1 kn + bn+1 kn−1 )hn − kn (an+1 hn + bn+1 hn−1 )
= bn+1 (kn−1 hn − kn hn−1 )
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
69. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Difference: Dn
hn
We wish to find the value of the continued fraction: lim .
n→∞ kn
n−1
h1 hn hr hr +1
Notice − = − , with this it becomes clear it
k1 kn kr kr +1
r =1
is important to define the difference: Dn+1 = kn+1 hn − kn hn+1 .
We have D1 = k1 h0 − k0 h1 = a1 a0 − 1 · (a1 a0 + b1 ) = −b1 and
Dn+1 = (an+1 kn + bn+1 kn−1 )hn − kn (an+1 hn + bn+1 hn−1 )
= bn+1 (kn−1 hn − kn hn−1 )
= −bn+1 Dn
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
70. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Difference: Dn
hn
We wish to find the value of the continued fraction: lim .
n→∞ kn
n−1
h1 hn hr hr +1
Notice − = − , with this it becomes clear it
k1 kn kr kr +1
r =1
is important to define the difference: Dn+1 = kn+1 hn − kn hn+1 .
We have D1 = k1 h0 − k0 h1 = a1 a0 − 1 · (a1 a0 + b1 ) = −b1 and
Dn+1 = (an+1 kn + bn+1 kn−1 )hn − kn (an+1 hn + bn+1 hn−1 )
= bn+1 (kn−1 hn − kn hn−1 )
= −bn+1 Dn
n
so any easy induction gives us Dn = (−1)n r =1 br .
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
71. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Summary: Dn
After I returned from the chalkboard, I used this slide.
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
72. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Summary: Dn
After I returned from the chalkboard, I used this slide.
To summarize what we just said:
Dn+1 = kn+1 hn − kn hn+1
n
Dn = (−1)n br
r =1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
73. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Telescoping Partials
Again, in the actual talk, I displayed “Hey Dan, go back to the
chalkboard please.” and just talked to them. I have included it
here since there is no chalkboard.
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
74. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Telescoping Partials
n
hn Dr
For the partials: = a0 −
kn kr kr −1
r =1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
75. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Telescoping Partials
n
hn Dr
For the partials: = a0 −
kn kr kr −1
r =1
n−1
h1 hn hr hr +1
− = −
k1 kn kr kr +1
r =1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
76. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Telescoping Partials
n
hn Dr
For the partials: = a0 −
kn kr kr −1
r =1
n−1
h1 hn hr hr +1
− = −
k1 kn kr kr +1
r =1
n−1
Dr +1
= but D1 = −b1 = a1 a0 − h1
kr kr +1
r =1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
77. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Telescoping Partials
n
hn Dr
For the partials: = a0 −
kn kr kr −1
r =1
n−1
h1 hn hr hr +1
− = −
k1 kn kr kr +1
r =1
n−1
Dr +1
= but D1 = −b1 = a1 a0 − h1
kr kr +1
r =1
n−1 n
hn h1 Dr +1 Dr
⇒ = − = a0 −
kn k1 kr kr +1 kr kr −1
r =1 r =1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
78. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Telescoping Partials
n
hn Dr
For the partials: = a0 −
kn kr kr −1
r =1
n−1
h1 hn hr hr +1
− = −
k1 kn kr kr +1
r =1
n−1
Dr +1
= but D1 = −b1 = a1 a0 − h1
kr kr +1
r =1
n−1 n
hn h1 Dr +1 Dr
⇒ = − = a0 −
kn k1 kr kr +1 kr kr −1
r =1 r =1
h1 a1 a0 −D1 a1 a0 D1
since k0 = 1, k1 = a1 ⇒ k1 = k1 = a1 − k1 k0 .
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
79. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Summary: Partials
After I returned from the chalkboard, I used this slide.
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
80. Intro
CF Difference
Convergents Telescoping Partials
Lemmata: Sums for π
Summary: Partials
After I returned from the chalkboard, I used this slide.
n
hn Dr
= a0 −
kn kr kr −1
r =1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
81. Intro
CF Review
Convergents Smith Sum
Lemmata: Sums for π
Introduction
The Wrong Way
Smart Men
Conway and Pell
Continued Fractions
Define It
What Does it Mean to Converge?
General
Working with the Convergents
Difference
Telescoping Partials
Lemmata: Sums for π
Review
Smith Sum
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
82. Intro
CF Review
Convergents Smith Sum
Lemmata: Sums for π
Review
Recall that we showed:
∞
π 1 1 1 (−1)n
= 1 − + − + ··· =
4 3 5 7 2n + 1
n=0
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
83. Intro
CF Review
Convergents Smith Sum
Lemmata: Sums for π
Smith Sum:
This is equivalent to the previous sum
Claim:
∞
π−3 (−1)n−1
=
4 (2n)(2n + 1)(2n + 2)
n=1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
84. Intro
CF Review
Convergents Smith Sum
Lemmata: Sums for π
Smith Sum:
This is equivalent to the previous sum
Claim:
∞
π−3 (−1)n−1
=
4 (2n)(2n + 1)(2n + 2)
n=1
Proof: Here we just manipulate the summand using the partial
fractal decomposition
1 1 1 1
= − + .
2n(2n + 1)(2n + 2) 4n 2n + 1 4n + 4
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
85. Intro
CF Review
Convergents Smith Sum
Lemmata: Sums for π
Smith Sum:
∞
(−1)n−1
2n(2n + 1)(2n + 2)
n=1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
86. Intro
CF Review
Convergents Smith Sum
Lemmata: Sums for π
Smith Sum:
∞
(−1)n−1
2n(2n + 1)(2n + 2)
n=1
∞ ∞ ∞
(−1)n−1 (−1)n (−1)n−1
= + +
4n 2n + 1 4n + 4
n=1 n=1 n=1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
87. Intro
CF Review
Convergents Smith Sum
Lemmata: Sums for π
Smith Sum:
∞
(−1)n−1
2n(2n + 1)(2n + 2)
n=1
∞ ∞ ∞
(−1)n−1 (−1)n (−1)n−1
= + +
4n 2n + 1 4n + 4
n=1 n=1 n=1
∞ ∞ ∞
(−1)n−1 (−1)n (−1)n
= + −1+
4n 2n + 1 4n
n=1 n=0 n=2
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
88. Intro
CF Review
Convergents Smith Sum
Lemmata: Sums for π
Smith Sum:
∞
(−1)n−1
2n(2n + 1)(2n + 2)
n=1
∞ ∞ ∞
(−1)n−1 (−1)n (−1)n−1
= + +
4n 2n + 1 4n + 4
n=1 n=1 n=1
∞ ∞ ∞
(−1)n−1 (−1)n (−1)n
= + −1+
4n 2n + 1 4n
n=1 n=0 n=2
(−1)1+1 π π−3
= + −1=
4·1 4 4
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
89. Analysis and Arithmetic
Parting Words
Part II
What We Came Here For: π
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
90. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Analysis and Arithmetic
A Nifty Identity
Denominator Series (kn )
Convergence
Plug and Play
Plug and Play
Parting Words
4 12
π and π 2
Thank You
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
91. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
A Nifty Identity
1 1 1 ∞ (−1)n−1
If c1 − c2 + c3 − ··· = n=1 cn converges then so does the
1
fraction 2
and they are
c1
c1 + 2
c2
c2 − c1 + 2
c3
c3 − c2 +
.
c4 − c3 + . .
equivalent, i.e. they are equal at every convergent.
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
92. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
A Nifty Identity
1 1 1 ∞ (−1)n−1
If c1 − c2 + c3 − ··· = n=1 cn converges then so does the
1
fraction 2
and they are
c1
c1 + 2
c2
c2 − c1 + 2
c3
c3 − c2 +
.
c4 − c3 + . .
equivalent, i.e. they are equal at every convergent.
Here a0 = 0, a1 = c1 , an = cn − cn−1 for n ≥ 2, b1 = 1, and
2
bn = cn−1 for n ≥ 2.
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
93. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Denominator Series
n
Claim: kn = i=1 ci .
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
94. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Denominator Series
Claim: kn = n ci .
i=1
Proof: We proceed by a strong induction. For n = 1, k1 = a1 = c1 ,
assuming it holds up to n = m for m ≥ 1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
95. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Denominator Series
Claim: kn = n ci .
i=1
Proof: We proceed by a strong induction. For n = 1, k1 = a1 = c1 ,
assuming it holds up to n = m for m ≥ 1
km+1 = am+1 km + bm+1 km−1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
96. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Denominator Series
Claim: kn = n ci .
i=1
Proof: We proceed by a strong induction. For n = 1, k1 = a1 = c1 ,
assuming it holds up to n = m for m ≥ 1
km+1 = am+1 km + bm+1 km−1
m m−1
2
= (cm+1 − cm ) ci + cm ci
i=1 i=1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
97. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Denominator Series
Claim: kn = n ci .
i=1
Proof: We proceed by a strong induction. For n = 1, k1 = a1 = c1 ,
assuming it holds up to n = m for m ≥ 1
km+1 = am+1 km + bm+1 km−1
m m−1
2
= (cm+1 − cm ) ci + cm ci
i=1 i=1
m m
= (cm+1 − cm ) ci + cm ci
i=1 i=1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
98. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Denominator Series
Claim: kn = n ci .
i=1
Proof: We proceed by a strong induction. For n = 1, k1 = a1 = c1 ,
assuming it holds up to n = m for m ≥ 1
km+1 = am+1 km + bm+1 km−1
m m−1
2
= (cm+1 − cm ) ci + cm ci
i=1 i=1
m m
= (cm+1 − cm ) ci + cm ci
i=1 i=1
m+1
= ci . So the induction holds.
i=1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
99. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Convergence
Claim:
∞
hn (−1)n−1
lim =
n→∞ kn cn
n=1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
100. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Convergence
Claim:
∞
hn (−1)n−1
lim =
n→∞ kn cn
n=1
n n−1 2
Proof: We have Dn = (−1)n i=1 bi = (−1)n i=1 ci . Given
what we just proved about kn :
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
101. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Convergence
Claim:
∞
hn (−1)n−1
lim =
n→∞ kn cn
n=1
n n−1 2
Proof: We have Dn = (−1)n i=1 bi = (−1)n i=1 ci . Given
what we just proved about kn :
n−1 2
Dn (−1)n i=1 ci (−1)n
= n n−1
=
kn kn−1 i=1 ci · i=1 ci
cn
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
102. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Convergence
Claim:
∞
hn (−1)n−1
lim =
n→∞ kn cn
n=1
n n−1 2
Proof: We have Dn = (−1)n i=1 bi = (−1)n i=1 ci . Given
what we just proved about kn :
n−1 2
Dn (−1)n i=1 ci (−1)n
= n n−1
=
kn kn−1 i=1 ci · i=1 ci
cn
∞
hn Dn
⇒ lim =0− since a0 = 0
n→∞ kn kn kn−1
n=1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
103. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Convergence
Claim:
∞
hn (−1)n−1
lim =
n→∞ kn cn
n=1
n n−1 2
Proof: We have Dn = (−1)n i=1 bi = (−1)n i=1 ci . Given
what we just proved about kn :
n−1 2
Dn (−1)n i=1 ci (−1)n
= n n−1
=
kn kn−1 i=1 ci · i=1 ci
cn
∞
hn Dn
⇒ lim =0− since a0 = 0
n→∞ kn kn kn−1
n=1
∞ ∞
(−1)n (−1)n−1
=− =
cn cn
n=1 n=1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
104. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Plug and Play
One final time in the actual talk, I displayed “Hey Dan, go back to
the chalkboard please.” and just talked to them. I have included it
here since there is no chalkboard.
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
105. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Plug and Play
We can use this identity in a surprising way to find the result of
this talk. The paper “An Elegant Continued Fraction for π” by
L.J. Lange credits one (presumably Douglas) D. Bowman for this
approach. Setting cn = 2n(2n + 1)(2n + 2) we have
a1 = c1 = 2 · 3 · 4 = 6(2 · 1)2 and for n ≥ 2,
an = 2n(2n + 1)(2n + 2) − (2n − 2)(2n − 1)2n = 24n2 = 6(2n)2 .
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
106. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Plug and Play
bn
In any general continued fraction, the step can be
bn+1
an +
..
.
cbn
transformed to for any c ∈ R× without changing the
cbn+1
can +
..
.
value of the expression.
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
107. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Plug and Play
bn
For n + 1 ≥ 2 we see the fraction =
bn+1
an +
..
.
(2n − 2)2 (2n − 1)2 (2n)2 (2n − 2)2 (2n − 1)2
= .
2+
(2n)2 (2n + 1)2 (2n + 2)2 (2n + 1)2 (2n + 2)2
6 · (2n) 6+
.. ..
. .
bn bn+1
So since bn+1 occurs in and , we cancel
bn+1 bn+2
an + an+1 +
.. ..
. .
(2n) 2 in the first and (2n + 2)2 in the second. After this reduction,
each an is simply 6 and each bn+1 becomes (2n + 1)2 for n ≥ 2.
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
108. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Plug and Play
Since b1 = 1 we are left to deal only with b2 . Our fraction has
become:
1 1
= .
22 32 42 4 · 32
6 · 22 + 6·4+
42 52 62 52
6 · 42 + 6+
. .
6 · 62 + . . 6 + ..
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
109. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Plug and Play
Since b1 = 1 we are left to deal only with b2 . Our fraction has
become:
1 1
= .
22 32 42 4 · 32
6 · 22 + 6·4+
42 52 62 52
6 · 42 + 6+
. .
6 · 62 + . . 6 + ..
And from our identity, this fraction is equal to
∞ ∞
(−1)n−1 (−1)n−1 π−3
= = .
cn 2n(2n + 1)(2n + 2) 4
n=1 n=1
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
110. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Plug and Play
This is the point where I returned from the chalkboard back to the
slides.
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
111. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Plug and Play
π−3 1
=
4 4 · 32
6·4+
52
6+
.
6 + ..
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
112. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Plug and Play
π−3 1 1 1
= =
4 4· 32 4 32
6·4+ 6+
52 52
6+ 6+
. .
6 + .. 6 + ..
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
113. A Nifty Identity
Denominator Series (kn )
Analysis and Arithmetic
Convergence
Parting Words
Plug and Play
Plug and Play
Plug and Play
π−3 1 1 1
= =
4 4· 32 4 32
6·4+ 6+
52 52
6+ 6+
. .
6 + .. 6 + ..
1
⇐⇒ π = 3 +
32
6+
52
6+
.
6 + ..
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series
114. Analysis and Arithmetic 4 and 12
π π2
Parting Words Thank You
Analysis and Arithmetic
A Nifty Identity
Denominator Series (kn )
Convergence
Plug and Play
Plug and Play
Parting Words
4 12
π and π 2
Thank You
Daniel J. Hermes Life of π: Continued Fractions and Infinite Series