Convergence of $\sum_{n = 1}^\infty \frac{(-1)^n}{n + n^2(1 + (-1)^n)}$

Question

Is $$\sum_{n = 1}^\infty \frac{(-1)^n}{n + n^2(1 + (-1)^n)}$$ convergent?

It is not absolutely convergent and Leibniz test is inconclusive.

Answer 1

Note that the series start with, $$ -1+\frac {1}{10} - \frac {1}{3} +...+ \frac {1}{2k+8k^2}-\frac {1}{2k+1} +...$$

$$ \frac {1}{2k+8k^2}-\frac {1}{2k+1} = \frac {1-8k^2}{(2k+8k^2)(2k+1)}$$

The series diverges by limit comparison test with $\sum {1/k}$

Answer 2

$$\begin{align} S= -\sum_{odd} \frac{1}{n} + \sum_{even} \frac{1}{n+2n^2}\\ \end{align}$$

The second series converges while the first doesn't; hence, the entire series diverges.