Does the series $\sum_{n\ge1}\left(-1\right)^n\frac{\cos\left(\alpha n\right)} {\sqrt n}$
converge ?
I tried to deal with this problem this way.
Let $S_k$ be a sequence of partial sums of the given series.
Than
$S_{2k}=\sum_{k=2}^{2n} \frac{\cos\left(\alpha n\right)} {\sqrt n}$,
series $\sum_{k=2}^\infty \frac{\cos\left(\alpha n\right)} {\sqrt n}$ converges by Dirichlet Convergence Test, therefore $S_{2k}$ converges.
And $S_{2k+1}=-\sum_{k=1}^{2n+1} \frac{\cos\left(\alpha n\right)} {\sqrt n}$.
If $S_{2k}\to 0$ than also $S_{2k+1}\to 0$ and the given series converges,
but I do not think that $S_{2k}$ must converge to zero.
Are there another approach to this problem ?
Thanks.
Best Answer
Answer. The series converges iff $a\ne (2k+1)\pi$.
This can be proved using Abel's summation method, since it is a series of the form $$ \sum_{n=1}^\infty a_nb_n $$ with $a_n=\frac{1}{\sqrt{n}}$ decreasing and tending to zero, and $b_n=(-1)^n\cos(an)$ which has bounded partial sums.