Differentiating infinite series

Question

Is this derivative correct? Does the index change when differentiating the series(increase by one)?

Given $f(x)=\sum_{k = 1}^{\infty} 2^kx^k$

Then $f'(x) = \sum_{k = 2}^{\infty}k2^kx^{k-1}$

and $\lim_{x\to0} f'(x) = 0$

Answer 1

That is not correct. If you look at the first term in $f(x)$, it is $2x$, after differentiating, it becomes $2$, but the first term in $\displaystyle\sum_{k=2}^{\infty}k2^{k}x^{k-1}$ is $2\cdot 2^{2}x$, which is different. The index should be preserved as $\displaystyle\sum_{k=1}^{\infty}k2^{k}x^{k-1}$.