power series

Question 1

1/x=

Answer 1

(-1)^n(x-1)^n
0<x<2

Question 2

1/1+x =

Answer 2

(-1)^nx^n
-1<x<1

Question 3

lnx=

Answer 3

(-1)^n-1(x-1)^n / n
0<x<=2

Question 4

e^x

Answer 4

x^n/n!
-inf < x < inf

Question 5

sinx

Answer 5

(-1)^n(x)^2n+1 / (2n+1)!
- inf. <x < inf

Question 6

cosx

Answer 6

(-1)^n (x)^2n / (2n!)
-inf <x< inf

Question 7

arctanx

Answer 7

(-1)^n x^2n+1 / 2n+1
-1<= x <= 1

Question 8

Rn(x)=

Answer 8

f^n+1 (2) / (n+1)! (x-c)^n+1)

Question 9

|Rn(x)|<

Answer 9

|x-c|^n+1/(n+1)! max |f^n+1 (z)|