AP® Calculus AB-BC

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Derivative of an Infinite Series

APCALC-T5UJNU

Suppose $f(x)$ is defined for $-1 < x < 1$ as follows:

$$f(x)=\sum _{ n=1 }^{ \infty }{ \frac { { \left( -1 \right) }^{ n }{ x }^{ 2n } }{ 2n } } $$

Find $f'(x).$

A

$$\sum _{ n=1 }^{ \infty }{ { \left( -1 \right) }^{ n } } { x }^{ 2n-1 }$$

B

$$\sum _{ n=1 }^{ \infty }{ { \left( -1 \right) }^{ n-1 } } { x }^{ 2n-1 }$$

C

$$\sum_{n=1}^{\infty }\frac{(-1)^{n}x^{2n+1}}{2n+1}$$

D

$$\sum_{n=1}^{\infty }\frac{(-1)^{n}x^{2n+1}}{2n(2n+1)}$$