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Determine for which real values of $\alpha$ and $q$ the following series converges $\sum\limits_{n=1}^{\infty}\frac{\cos(\alpha\sqrt{n})}{n^q}$?
So far I managed to prove that 1) for $q\leqslant0,\alpha\in\mathbb{R}$ the series diverges; 2) for $q>1,\alpha\in\mathbb{R}$ the series converges absolutely; 3) for $0<q\leqslant 1,\alpha=0$ the series diverges.

How to approach the problem for the case $0<q\leqslant 1,\alpha\neq0$?

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  • $\begingroup$ The series diverges for $q \in (0,1/2]$ and $\alpha \in \mathbb{R}$. The idea is that you can find intervals of length $\sim \sqrt{n}$ such that $\cos (\alpha \sqrt{n}) \geq 1/2$, so the sum of the sequence along such an interval is roughly $\sum_{k=n}^{n+\sqrt{n}} k^{-q} \sim n^{\frac{1}{2}-q}$. My guess is that the series converges for $q > 1/2$, $\alpha \neq 0$ via some kind of "alternating sequence" criterion, but I don't have any definite argument on top of my head. $\endgroup$ – D. Thomine Dec 17 '15 at 11:21
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By the Euler-MacLaurin formula, we have

$$S(k) := \sum_{n = 1}^k \cos (\alpha\sqrt{n}) = \int_1^k \cos (\alpha \sqrt{t})\,dt + O(1).$$

Further,

\begin{align} \int_1^k \cos (\alpha \sqrt{t})\,dt &= 2\int_1^{\sqrt{k}} u\cos (\alpha u)\,du\\ &= \frac{2}{\alpha}\bigl[u\sin (\alpha u)\bigr]_1^{\sqrt{k}} - \frac{2}{\alpha}\int_1^{\sqrt{k}} \sin (\alpha u)\,du \end{align}

shows $\lvert S(k)\rvert \leqslant \frac{4}{\lvert\alpha\rvert}\sqrt{k} + O(1) \leqslant C\sqrt{k}$ for some constant $C \in (0,+\infty)$.

Then a summation by part shows

\begin{align} \Biggl\lvert\sum_{n = m}^k \frac{\cos (\alpha \sqrt{n})}{n^q}\Biggr\rvert &= \Biggl\lvert\sum_{n = m}^k \bigl(S(n) - S(n-1)\bigr) n^{-q}\Biggr\rvert\\ &\leqslant \frac{\lvert S(k)\rvert}{k^q} + \frac{\lvert S(m-1)\rvert}{m^q} + \Biggl\lvert \sum_{n = m}^{k-1} S(n) \biggl(\frac{1}{n^q} - \frac{1}{(n+1)^q}\biggr)\Biggr\rvert\\ &\leqslant C\Biggl( k^{\frac{1}{2}-q} + m^{\frac{1}{2}-q} + \sum_{n = m}^{k-1}\frac{q}{n^{q+\frac{1}{2}}}\Biggr)\\ &\leqslant \tilde{C}\cdot m^{\frac{1}{2}-q} \end{align}

for $q > \frac{1}{2}$.

So as D. Thomine expected, the series converges for $q > \frac{1}{2}$ and $\alpha \in \mathbb{R}\setminus \{0\}$. The argument given in the comment shows (when the details are carried out) that the series is divergent for $q \leqslant \frac{1}{2}$.

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