$\text { Consider the function given by } f ( x ) = \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } ( x - 6 ) ^ { n } } { n } \text {. Find the interval of }$ convergence for $f ^ { \prime } ( x )$.
A) $[ - 6,6 ]$
B) $[ 5,7 ]$
C) $( 0,7 )$
D) $( - 6,6 )$
E) $( 5,7 )$

Question

Quizplus · Accepted Answer

The derivative of the given function, $f'(x)$, is found by differentiating term by term, which results in a geometric series. The interval of convergence for $f'(x)$ is the same as the interval of convergence for the original series, which is determined by the ratio test. The original series converges for $|x-6|<1$, leading to the interval $(5,7)$ for convergence.

Convergence For f′(x)f ^ { \prime } ( x )f′(x)

Convergence For $f ^ { \prime } ( x )$