Find the sum of this series $\sum_{k=1}^{\infty} \frac{(2-x)^k}{2^k*k}$ How to find the integration $C$?

Question

Hello I have the following series :

$$\sum_{k=1}^{\infty} \frac{(2-x)^k}{2^k*k}$$

I found that the series convergents for $0<x \leq 4$

I managed to reach that

$$f(x)=-ln(x)+C$$

But for some reason the answer is that $$f(x)=ln(\frac{2}{x})$$

I don't understand how they found $C$?

Any ideas, how $C$ could be found?

Thank you!

Answer 1

If you insert $x=2$ you get that the series equals zero, since all terms equal zero. That gives you $$ 0=f(2)=-\ln 2+C $$ and hence $$ C=\ln 2. $$