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.
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.
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 $α=0.5$.