3
$\begingroup$

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.

  • 1
    @ Zev Chonoles: I will start reading how to format mathematics. Thank you.2012-12-23

2 Answers 2

2

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 $α=0.5$.

1

Use Dirichlet test for series convergence.

  • 0
    @Downvoter: What's the down for?2012-12-25