Product of series converging?

Question

$$\sum_{k=1}^\infty (-1)^k \left(1 - \frac 1 {{3k}}\right) ^{k} $$

So the first part is the geometric series which diverges..

How do I deal with such series?

Answer 1

Recall that

$$S_n=\sum_{k=1}^n a_n \to L\in \mathbb{R} \implies S_{n}-S_{n-1}=a_{n} \to 0$$

therefore $a_n \to 0$ is a necessary condition (but not sufficient) for convergence and as noticed here we have

$$\left(1 - \frac 1 {{3k}}\right) ^{k}=\left[\left(1 - \frac 1 {{3k}}\right) ^{3k}\right]^{1/3}\to \frac1{\sqrt[3]e}$$

Answer 2

Hint: What can you say about the limit $\displaystyle\lim_{k\to\infty}\left\lvert(-1)^k\left(1-\frac1{3k}\right)^k\right\rvert$?