Mathematics

Problem

Let

g(x)=\displaystyle \sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n}(x-2)^n

, which converges for

|x-2|<1

. Which power series represents

g'(x)

? a)

\displaystyle \sum_{k=0}^{\infty} \dfrac{(-1)^{k}}{k+1}(x-2)^{k}

\displaystyle \sum_{k=1}^{\infty} (-1)^{k-1}(x-2)^{k}

\displaystyle \sum_{k=0}^{\infty} (-1)^{k}(x-2)^{k}

\displaystyle \sum_{k=0}^{\infty} (-1)^{k}(k+1)(x-2)^{k}