calculus - Convergence of $sum_{n=0}^infty frac{cos(n)}{n}$

Does this series converge

$$\sum_{n=1}^\infty \frac{\cos(n)}{n}$$

someone has told me that I have to apply Dirichlet's test but I don't know how to calculate the sum

$$\left|\sum_{n=1}^{N} \cos(n)\right|$$

Answer

It is the finite sums that you have to bound (it does not imply that the series converges). You have

$$\begin{align} \left|\sum_{n=1}^N\cos n\right| & =\left|\sum_{n=1}^N \operatorname{Re}\,e^{in}\right| =\left|\operatorname{Re}\,\sum_{n=1}^N e^{in}\right| =\left|\operatorname{Re}\,\frac{e^{i}-e^{i(N+1)}}{1-e^i}\right| \\[10pt] & \leq\left|\frac{e^{i}-e^{i(N+1)}}{1-e^i}\right| \leq\frac2{|1-e^i|} =\frac2{\sqrt{(1-\cos1)^2+\sin^21}}=\frac{\sqrt 2 }{\sqrt{1-\cos1}} \end{align}$$