The convergence or the divergence of a series

Question

Let us consider the series of general term:

$$\frac{(-1)^{n-1}}{n^{1/2}}\sin(\beta \log n)$$

The question is about the convergence or the divergence of this series.

Answer 1

We can use the complex series of the general term: $$\frac{(-1)^{n-1}}{n^{1/2}}\exp(-\beta \log n)$$ to obtain the the eta function which is analytic in the domain $Re(α+iβ)>0$. The mentioned series in the question is the imaginary part of the eta function which is convergent since the whole series is convergent. Here we have $α=0.5$.

Answer 2

Use Dirichlet test for series convergence.