The document introduces the integral test, which can be used to determine if an infinite series is convergent or divergent. The integral test states that a series is convergent if the corresponding improper integral converges, and divergent if the improper integral diverges. Several examples demonstrate applying the integral test to various series by setting up and evaluating the corresponding improper integrals. The document also discusses using integrals to estimate the sum of a convergent series.

1. 11.3 The Integral Test
There are many series that cannot be easily
evaluated. We need a method to
determine if it is convergent without
knowing the precise quantity.
estimate the sum approximately.
Ex:
=
+

2. The integral test:
Suppose ( ) is a continuous positive
decreasing function and let = ( ) , then
the series
=
is convergent if and only if the improper
integral
( )
is convergent.

3. Improper integral of Type I:
If ( ) exists for every , then
( ) = ( )
provided this limit exists as a finite number.
We call this improper integral convergent,
otherwise divergent.
a ∞

4. Ex: Determine if series is convergent.
=
+

5. Ex: Determine if series is convergent.
=
+
Let ( )=
+
it is continuous, positive and decreasing.

6. Ex: Determine if series is convergent.
=
+
Let ( )=
+
it is continuous, positive and decreasing.
=
+ +

7. Ex: Determine if series is convergent.
=
+
Let ( )=
+
it is continuous, positive and decreasing.
=
+ +
=

8. Ex: Determine if series is convergent.
=
+
Let ( )=
+
it is continuous, positive and decreasing.
=
+ +
=
=

9. Ex: Determine if series is convergent.
=
+
Let ( )=
+
it is continuous, positive and decreasing.
=
+ +
=
=
So is convergent.
=
+

10. Ex: Determine if series is convergent.
=
+
Let ( )=
+
it is continuous, positive and decreasing.
=
+ +
=
=
it is!
So is convergent. what
+ know
don’t
ut we
=
B

11. Ex: Determine if series is convergent.
=

12. Ex: Determine if series is convergent.
=
Let ( )=

13. Ex: Determine if series is convergent.
=
Let ( )=
it is continuous, positive but not decreasing.

14. Ex: Determine if series is convergent.
=
Let ( )=
it is continuous, positive but not decreasing.
however, it is decreasing when > , fine.

15. Ex: Determine if series is convergent.
=
Let ( )=
it is continuous, positive but not decreasing.
however, it is decreasing when > , fine.
=

16. Ex: Determine if series is convergent.
=
Let ( )=
it is continuous, positive but not decreasing.
however, it is decreasing when > , fine.
=
( )
= =

17. Ex: Determine if series is convergent.
=
Let ( )=
it is continuous, positive but not decreasing.
however, it is decreasing when > , fine.
=
( )
= =
So is divergent.
=

18. Ex: p-test
is convergent if > , otherwise divergent.

19. Ex: p-test
is convergent if > , otherwise divergent.
is convergent if > , otherwise divergent.
=

20. Reminder estimate:
Suppose ( ) is a continuous positive
decreasing function and let = ( ).
∞
If = = is convergent, then
+ ( ) + ( )
+
or
( ) ( )
+
where = is called the remainder.

21. Ex: Estimate the sum of the series
=
using = .

22. Ex: Estimate the sum of the series
=
using = .
( )= is continuous positive decreasing

23. Ex: Estimate the sum of the series
=
using = .
( )= is continuous positive decreasing
+ +

24. Ex: Estimate the sum of the series
=
using = .
( )= is continuous positive decreasing
+ +
+ +
· ·

25. Ex: Estimate the sum of the series
=
using = .
( )= is continuous positive decreasing
+ +
+ +
· ·
= + + ··· + .

26. Ex: Estimate the sum of the series
=
using = .
( )= is continuous positive decreasing
+ +
+ +
· ·
= + + ··· + .
. .