# Why : $\sum\limits_{k=1}^\infty \frac{\sin(k)}{\sqrt{k}},\sum\limits_{k=1}^\infty \frac{\cos(k)}{\sqrt{k}}$ Converges? [closed]

if we look to those trigonometric Sum : $$\sum\limits_{k=1}^\infty \frac{\sin(k)}{\sqrt{k}},\sum\limits_{k=1}^\infty \frac{\cos(k)}{\sqrt{k}}$$ one can say they could be diverge since $$\sin$$ and $$\cos$$ functions are bounded and lie in $$[-1;1]$$ , one can get two divergent sum if we take $$\sin(k)$$ is $$1$$ for $$k \to \infty$$ so we have $$\sum_{k\geq1} \frac{1}{\sqrt{k}}$$ which is diverge the same idea with $$\cos$$ function, Now i want if there is any theorem with proof show why exactly the titled sum are conveges ? and what is its geometric interpretation ?

## closed as off-topic by Zacky, Cesareo, Eevee Trainer, Song, LeucippusMar 15 at 4:17

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• Because of Dirichlet's test and this. – rtybase Mar 14 at 22:34
• For $x \mathbb{R}, \not \in 2\pi \mathbb{Z}$ then $\sum_{k=1}^\infty e^{ik x} k^{-s} = \sum_{k=1}^\infty (\sum_{n=1}^k e^{inx}) (k^{-s}-(k+1)^{-s}) = \sum_{k=1}^\infty \frac{1-e^{ikx}}{1-e^{ix}} \int_0^1 s (k+t)^{-s-1}dt$ – reuns Mar 14 at 22:52