Limit and a series $$\lim_{n\to+\infty}⁡\frac{\cos⁡(\frac1n)+\cos⁡(\frac2n)+⋯+\cos⁡(\frac nn)}{n}$$
What I have tried:
I tried rewriting the series portion as the sum $i=1$ to n of $\cos(i/n)$. Then used the formula the sum $i=1$ to n of i = n(n+1)/2 and substituted for i. Then i had the lim(n→∞)⁡[(1/n)cos(n(n+1)/(2n)}] which would be just 0

We have
$$\lim_{n\to \infty} \frac{1}{n}\sum_{j=1}^n \cos \frac{j}{n}= \int_0^1 \cos x \;\mathrm d x = \sin 1.$$
