I am looking for some help with an real analysis problem that I have.
Problem: Find the sum of the series $\sum_{n=1}^{\infty}(-1)^{n+1}(\frac{1}{n})=1-(1/2)+(1/3)-(1/4)+\ldots $
What I have so far: My intuition suggests that I could use Abel's theorem which states that if $G(x)=\sum_{k=0}^{\infty}a_{k}x^{k}$ is a power series with real coefficients converges and the radius of convergence is 1, then $\lim_{x\to1^{-}}G(x)=\sum_{k=0}^{\infty}a_k$.
So then I tried to rewrite the power series given in the problem to have a starting index of 0. $\sum_{n=1}^{\infty}(-1)^{n+1}(\frac{1}{n})=\sum_{k=0}^{\infty}(-1)^{k}\frac{1}{k+1}=\lim_{x\to1^{-}}\ln(1+x)=ln(2)$ which I think would hold because the radius of convergence of $\ln(1+x)$ is 1. I am not sure if this is the right idea or not.