# Uniform convergence of $\sum_{k=2}^{\infty}\left(\cos\frac{x}{k}-\cos\frac{x}{k-1}\right)$

I tried to use Weierstrass M-test for checking if $$\sum_{k=2}^{\infty}\left(\cos\frac{x}{k}-\cos\frac{x}{k-1}\right)$$ converges uniformly on $$(-\infty, \infty)$$ and I got $$\left|\cos\frac{x}{k}-\cos\frac{x}{k-1}\right|\leq\left|\cos\frac{x}{k}\right|+\left|\cos\frac{x}{k-1}\right|\leq1+1=2$$,

$$\sum_{k=2}^{\infty}2$$ diverges $$\Rightarrow \sum_{k=2}^{\infty}\left(\cos\frac{x}{k}-\cos\frac{x}{k-1}\right)$$ doesn't converge uniformly on $$(-\infty, \infty)$$.

Is this correct?

• You have shown that your series is less than a divergent series. That is not enough to show convergence – whpowell96 May 24 '20 at 21:03
• This is a telescoping series. Use that to obtain the partial sums in closed form. This will tell you whether the series converges and whether the convergence is uniform for all $x$. – Hans Engler May 24 '20 at 21:05

$$\sum_{k=2}^n \cos\left(\frac{x}{k}\right) - cos\left(\frac{x}{k-1}\right) = cos\left(\frac{x}{n}\right) - \cos x \to 1 - \cos x$$