Hello, I need to investigate Uniform convergence for these two using Weierstrass M-test, 10x in advance.
Assaf.
For the first problem note $\displaystyle x^n + x^{-n} \leq 2$.
Thus, $\displaystyle |f_n(x)| \leq \frac{2n^2}{\sqrt{n!}}$ and $\displaystyle \sum_{n=0}^{\infty} \frac{2n^2}{\sqrt{n!}} < \infty$.
I did not do the second problem but I would imagine you need the fact $\displaystyle \log \left( 1 + \frac{x^2}{n\log^2 n} \right) \leq \log \left( 1+\frac{a^2}{n\log^2 n} \right)$.
Now let us hope that $\displaystyle \sum_{n=2}^{\infty} \log \left( 1+\frac{a^2}{n\log^2 n} \right) < \infty$.
It is! And there is an easy way to show it.
You need the fact that $\displaystyle \log (1+x) \leq x$ for $\displaystyle x\geq 0$.
Thus,
$\displaystyle 0\leq \log \left( 1 + \frac{a^2}{n^2\log n} \right) \leq \frac{a^2}{n^2\log n}$
And of course, by applying Betrand's test
$\displaystyle \sum_{n=2}^{\infty}\frac{a^2}{n^2\log n} < \infty$