1. I. Escribir los Desarrollos en serie de Potencias de “x” e indicar los intervalos de
convergencia de las siguientes funciones:
1. f(x) =
)x21)(x1(
3

; 




0n
n1nn
]x.21)([1 ; x < ½
2. f(x) = 2
)1x(
)3x2(


; - 



0n
n
3)x(n ; x < 1
3. f(x) = x. x2
e
; x + 



2n
n1n-1n
1)!(n
x2(-1)
; x < 
4. f(x) =
2
x
e ; /n!x1
1n
2n



 ; x < 
5. f(x) = Senh(x) ; 



0n
12n
1)!(2n
x
; x < 
6. f(x) = Cos(2x) ; 



1n
2n2n
n
(2n)!
.x2
1)(1 ; x < 
7. f(x) = Cos2
(x) ; 




1n
2nn
(2n)!
(2x)1)(
1/21 ; x < 
8. f(x) = 2
x9
x

; 





0n
1n
12n
n
9
x
1)( ; x < 3
9. f(x) =
2
x4
1

; 12n
2n
0n 2
x
2.4.6...2n
1)2n1.3.5....(





; x < 2
10. f(x) = Ln 







x1
x1
; 



0n
12n
12n
x
2 ; x < 1
11. f(x) = ax
(a > 0) ; 



1n
nn
n!
.a.Lnx
1 ; x < 
12. f(x) = Ln(2 + x) ; Ln(2) + x/2 + x2
/2.22
+ x3
/3.23
+ … + n
n
1n
2n
x
.)1( 
 ; 2 < x  2
13. f(x) = Sen2
(x) ; 





1n
2n12n
1n
(2n)!
.x2
1)( ; x < 

2. 14. f(x) = Cos(x + a) ; Cos(a) – xSen(a) – x2
.Cos(a) + x3
Sen(a) + x4
Cos(a) + .... x < 1
2! 3! 4!
15. f(x) = Ln(x + 2
x1  ) ;
12n
x
n)2.4.6...(2
1)n1.3.5...(2
1)(
12n
n
n





 ; x  1
16. f(x) = Arctg(x) ; 12n
1n
1n
x
12n
1)( 



 

; [-1, 1]
17. f(x) = Sen(x).Cos(x)
18. f(x) = 3
x8 
19. f(x) =
x
)x(Cos1 
20. f(x) =
2
x1
1

21. f(x) = x2
.ex
22. f(x) = x2
Sen(x)
23. f(x) = Cos(x2
)
24. f(x) = x(1 + 2x)-2
II. Calcular el Límite, Usando Series de Potencias:
1.
)1x(Lnx
)x(Arctg)x(Sen
Lim 20x 


=
6
1
2.
)x(Arctgx
x2ee
Lim
xx
0x 
 

= 1
3.
)1x(Lnx
x)x(Arctg
Lim 20x 


=
3
1

4. 







 2
2
20x x1
x1
Ln.
x
1
Lim = 2
5. 







 1x
1x
SenLim
2
1x
= 2
6.
)x(Senx
x)x(Tg
Lim
0x 


= 2
7.
)x(Arctg
24x2
Lim
0x

 2
1

8. 20x x
))x(Cos(Ln
Lim

=
2
1

9. 30x x
)x(Senh)x(Tg
Lim


=
6
1
10.
)1e(x
)x(Arcsen)x1(Ln
Lim x0x 


=
2
1

11.   3
2
xxxxLim 3 233 23
x



3. 12.
)1x(Lnx
)x(Arctg)x(Sen
Lim 20x 


=
6
1
13. 30x x
)x(Sec)x(Tg
Lim


= -
14. 20x ))x(Cosh1(
)x(Senhx
Lim



=
15.
20
23
x
exx3))x(Cos2)(x(Sen
Lim 5
x3
0x



16.
)2/x(Sen
)x(Cose
Lim
x
0x


=
17.
)x(Sen
)x(Arcsenx
Lim 30x


=
6
1

18.
416x
39x
Lim
2
2
0x



=
3
4
19.
)x(Sen)x1(Ln
1)x1(
Lim
2/1
0x 


=
20.
28x2
x)x(Arcsen
Lim
3 20x



=
21. )x(Cotg)x(CoscLim
0x


= 0
22. 3
2
0x x
)xx1(Lnx
Lim


=
6
1
23. 






 )x(Cos
1
1x
1
Lim
0x
=
24.
x1
x1
Lim
1x 


=
25. 1
)1e(x
)xx1(Ln)xx1(Ln
Lim x
22
0x




26. 4
)x(Cos)x(Cos
)x(Sen2)x(Sen2)x2(Sen
Lim 2
2
0x




27.
)x(xSen
)x(Cos1
Lim
0x


=
2
1
28.
x/1
)x/1(Arctg2x/1
Lim
x


= -1
29. 






 x
)x(Cotg
x
1
Lim 20x
=
3
1
30.
xx
)x(Tg3)x(Sen
Lim 3
2
0x 


=
31. 3x ))x(Tg(
)x(Arctgx
Lim


=
32. 20x x1
))x(Sen1(Ln
Lim



=

4. x + _1_.x5
+ _1.3__.x9
+ … + 1. 3.5…(2n – 1)x4n +1
x < 1
2.5 22.
9.2! 2n
(4n + 1)n!
III. Calcular las siguientes integrales utilizando desarrollos en series de potencias e
indicar los intervalos de Convergencia:
1. 
t
0
x
sen(x)dx
;
1)!1)(2n(2n
x
1)(
12n
0n
n




 ; x < 
2. dxe
x
0
x2


; 





1n
12n
n
1)n!(2n
x
1)(x ; x < 
3.  
x
0
x)/x)dx(Ln(1 ; 




1n
2
n
1n
n
x
1)( ; x  1
4.  
x
0
4
x1
dx
;
5. 
x
0
Arctg(x)
6. dx)xLn(1
x
0
2
 
IV. Resolver las siguientes Integrales:
1. f(x) = t)dt/(1e
x
0
t
 ; x + x3
/6 – x4
/12 + 3x5
/40 - ….
2. f(x) = 
x
0
/tArctg(t)dt
3. f(x) = 2
x
0
t
1dt/te
2

4. f(x) = )dtt.Cos(e
x
0
t


5. f(x) = dt))tLn(Cos(
x
0


5. V. Usar una serie de Potencia, encontrar el valor aproximado con una exactitud de 4
cifras decimales:
1. dxe
1
0
2x


; R = 0.7468
2. 

1
0
2
dx
x
Cos(x)1
R = 0.4864
3. dx
x
x)Ln(1
1/2
0


R = 0.4484
4. 
1/2
0
dx
x
Arctg(x)
R =
5. dx
x
e1
1/2
0
x



R =
6. dx)Sen(x
1/2
0
2
 R = 0.0415
7.  
1/2
0
3
)xdx/(1 R = 0.04854
8. dx
x
Sen(x)
1
0
 R = 0.621
9. 
1
0
3
Cos(x)dxx R = 0.608
10. dxx1
1/2
0
3
  R = 0.508
11. dxex
2
x
1
0
2 
 R = 0.189
12. dxxArctg(x)
0.3
0
 R = 0.0088
13. 
1/2
0
2
)dxArctg(x R =
14. 
1
0
4
)dxx.Sen(x R =
VI. Calcular el valor aproximado usando series de Potencias:
1. 24 ; R = 4.899
2. 5
30 ; R =1.974
3. 102 ; R = 10.09995

6. VII.