Show that the given series in Uniformly convergent

Question

Show that the series $$1+{e^{-2x}\over2^2-1}-{e^{-4x}\over4^2-1}+{e^{-6x}\over6^2-1}-\cdot\cdot\cdot$$

is uniformly convergent for all real $x\ge0.$

I tried applying the Dirichlet's theorem to the problem. The given series is $$S(x)=\sum_{n=0}^{\infty} {(-1)^{n-1}e^{-2nx}\over (2n)^2-1}$$

The summation $S(x)$ can be shown to be bounded. Also, for a fixed $x\geq 0,$ the sequence $\displaystyle \frac{e^{-2nx}}{ (2n)^2-1}$ is positive and monotonically decreasing. How do I show that this sequence converges uniformly to $0$ ?

I need some clues.

Answer 1

The key here is the Weierstrass M-test:

Let $D\subseteq \mathbb{R}$ and $f_n:D\to \mathbb{R}.$ Suppose that for each $f_n,$ there exists $M_n\in \mathbb{R}$ such that $| f_n (x) | \leq M_n$ for all $x\in D$ and $\sum_{n=0}^{\infty} M_n < \infty.$ Then $\sum_{n=0}^{\infty} f_n(x)$ converges uniformly on $D.$

Proof: Let $\epsilon>0.$ Since $\sum M_n$ converges, there exists an $N$ such that $\sum_{k=N+1}^{\infty} M_k < \epsilon.$ Now $$\sup_{x\in D} \left| \sum_{k=n+1}^{\infty} f_k(x) \right| \leq \sum_{k=n+1}^{\infty} \sup_{x\in D} |f_k(x)| \leq \sum_{k=n+1}^{\infty} M_k < \epsilon.$$ Thus the series converges uniformly on $D.$

Now can you find such $M_n$ for your series?