# Uniform convergence on $\mathbb R$ of the series $\sum_{n=2}^{\infty} \frac{(-1)^{n+1}} {\sqrt n + \cos x}$

Is the following series of functions uniformly convergent on $$\mathbb{R}$$?

$$\sum_{n=2}^{\infty} \frac{(-1)^{n+1}} {\sqrt n + \cos x}$$

My attempt: My answer is No

I know that by Leibnitz test the given series converges, but by using $$M$$- test

$$\sup_{x \in \mathbb{R}}\left|\sum_{n=2}^{\infty} \frac{(-1)^{n+1}} {\sqrt n + \cos x}\right| \le \sup_{x \in \mathbb{R}} \left|\sum\frac{1}{\sqrt n}\right| \neq 0$$

so the given series is not uniformly convergent on $$\mathbb{R}$$

Is it correct ?

• Why is the inequality after "My attempt" true? If it were true, how would it allow you to answer the question? – Did Sep 23 '18 at 9:14

Yes, it converges uniformly on $$\mathbb R$$. You just apply Dirichlet's test for uniform convergence:
1. $$\sum_{n=0}^N(-1)^n$$ is uniformly bounded;
2. for each real $$x$$, $$\left(\dfrac1{\sqrt n+\cos(x)}\right)_{n\in\mathbb N}$$ is monotonic;
3. $$\left(\dfrac1{\sqrt n+\cos(x)}\right)_{n\in\mathbb N}$$ converges uniformly to $$0$$.
Simply use the usual bound on the remainder of an alternating series $$\left|\sum_{k=n}^{\infty} \frac{(-1)^{k+1}} {\sqrt k + \cos x}\right|\leq \frac{1}{\sqrt n +\cos(x)}\leq \frac{1}{\sqrt n-1}$$
Since the bound is independent of $$x$$ and goes to $$0$$, there is indeed uniform convergence.