Explore $\sum_\limits{n = 1}^{\infty} \frac{\cos n \cdot \cos( 1/n)}{\sqrt[4] n}$ Explore absolute and conditional convergence $\sum_\limits{n = 1}^{\infty} \frac{\cos n \cdot \cos (1/n)}{\sqrt[4] n}$
How to bound $|\frac{\cos n \cdot \cos (1/n)}{\sqrt[4] n}|$?
For conditional I tried:
$$\sum_\limits{n = 1}^{\infty}\frac{\cos n}{\sqrt[4]{n}}-\sum_\limits{n = 1}^{\infty}\frac{\cos n\cdot\Big(1-\cos\frac{1}{n}\Big)}{\sqrt[4]{n}}$$
I want to use sign of Dirichlet
Why partial sum of $\cos n$ is limited?
 A: (1) For the absolute convergence, notice that for each $k = 1, 2, \cdots$ there is at least one $n$ such that $n \in [k\pi - \frac{1}{2}, k\pi + \frac{1}{2}] =: I_k$. Then for such $n$, we have
$$ \left| \frac{\cos n \cos(1/n)}{n^{1/4}} \right| \geq \frac{\cos(\frac{1}{2})\cos 1}{(k\pi + \frac{1}{2})^{1/4}} \geq \frac{c}{k^{1/4}}$$
for some generic constant $c > 0$. (For instance, $c = \cos(\frac{1}{2})\cos(1)/(\pi+\frac{1}{2})^{1/4}$ works.) Since $I_j$ and $I_k$ are disjoint if $j \neq k$, it follows that
$$ \sum_{n=1}^{\infty} \left| \frac{\cos n \cos(1/n)}{n^{1/4}} \right| \geq \sum_{k=1}^{\infty} \frac{c}{k^{1/4}} = \infty. $$
(2) The partial sum of $\cos n$ is bounded by the following computation:
$$ \left| \sum_{n=1}^{N} \cos n \right|
= \left| \Re\bigg( \sum_{n=1}^{N} e^{in} \bigg) \right|
= \left| \Re\bigg( \frac{e^i}{1-e^i}(1 - e^{Ni}) \bigg) \right|
\leq \frac{2}{|1 - e^{i}|}. $$

(1') A better estimate can be obtained by utilizing certain averaging technique. Let
$$s_n = \sum_{k=1}^{n} \left|\cos k\right|\left|\cos(1/k)\right|.$$
Using the Weyl's criterion, we can check that
$$ \frac{s_n}{n} \sim \int_{0}^{1} \left|\cos(\pi x)\right| \, dx = \frac{2}{\pi}. $$
So we expect that the overall behavior of the partial sum remains the same if we replace $\left|\cos n \cos(1/n)\right|$ by its average $\frac{2}{\pi}$:
$$ \sum_{n=1}^{N} \frac{\left|\cos n \cos(1/n)\right|}{n^{1/4}} \sim\sum_{n=1}^{N} \frac{2}{\pi}\frac{1}{n^{1/4}} \sim \frac{8}{3\pi}N^{3/4}.$$
This heuristics can be made rigorously by using summation by parts technique.
