Does $ \sum_{n=1}^{\infty} (-1)^n\frac{3n-2}{n+1}\frac{1}{n^{1/2}}$ converge?

Question

I think that there could be used Abel and Dirichlet method, but I have no idea how

$$ \sum_{n=1}^{\infty} (-1)^n\frac{3n-2}{n+1}\frac{1}{n^{1/2}} .$$

Answer 1

The series $$\sum_{n=1}^\infty\frac{3(-1)^n}{n^{1/2}}$$ is convergent by Leibniz. The difference from the original series is $$\sum_{n=1}^\infty (-1)^n\left(\frac{3n-2}{(n+1) n^{1/2}}-\frac{3}{n^{1/2}}\right)$$ which is aboslutely convergent, since the terms are $O(n^{-3/2})$.

Answer 2

Write the fraction as $3-5/(n+1)$.

The sum of the first converges by Leibnitz and the second converges absolutely.