Exercise 37 in Apostol $10.20$ asks to find all complex $z$ such that $\sum_{n=1}^{\infty} \frac{(-1)^n}{z+n}$ converges.
I suspect this requires the use of either Abel's Test or Dirichlet's Test. My attempt so far has been to set $\{b_n\}=\frac{1}{n}$, which is a decreasing sequence of real numbers that converges to $0$. Then, I set $\{a_n\}=(-1)^n\frac{n}{z+n}$.
As $n\to\infty$, $\{a_n\} \to (-1)^ne^{i\arg(\frac{n}{z+n})}$ since $|\frac{n}{z+n}|=\frac{n}{|z+n|}\to 1$
In order to prove that this converges for all complex $z\not=-1,-2,\dots$, I must show that $A_n=\sum_{k=1}^{n} a_n$ is a bounded sequence (not necessarily that it converges). This would satisfy the hypotheses for Dirichlet's test.