Does the alternating series $\sum_{n=1}^{\infty}(-1)^n$ converge?

Question

Does the series

$$\sum_{n=1}^\infty (-1)^n $$

converge? I was trying to use this as a convergent majorant for proving convergence of $$\sum_{n=1}^\infty (-1)^n \frac {\ln(n)} n $$ but I'm not sure if that would work. Any help is appreciated!

Answer 1

Think about what $\sum_{n=1}^{\infty}(-1)^{n}$ means

You get $-1 + 1 -1 + 1...$

Does that converge to a number? Clearly not, it goes back and forth between $-1$ and $0$

A convergent sum will get closer and closer to one particular value, and can in fact get arbitrarily close to that value, which clearly our sum does not.

Answer 2

Given $n\geq 1$, you have that $\frac{\ln n}{n}\geq 0$. Moreover, $n\mapsto \frac{\ln n}{n}$ is a decreasing function, which tends to $0$ as $n\to\infty$. Thus, you may apply the Alternating Series Test to conclude that the series $\sum_{n=1}^{\infty}(-1)^{n}\frac{\ln n}{n}$ converges.