Uniform convergence by Abel's test

Question

Using Abel's test on uniform convergence, how to examine the uniform convergence of $\sum\displaystyle\frac{(-1)^n(1+|\cos^nx|)}{x+n}$ on $[0,\infty)$?

Answer 1

1. Show $\displaystyle\sum\frac{(-1)^n}{x+n}$ converges uniformly on $[0,\infty)$ (you may want to use Dirichlet's test.)
2. Show $1+|\cos^nx\,|$ is uniformly bounded.
3. Show $1+|\cos^nx\,|$ is monotone for all $x\ge0$.