# Math Help - Uniform convergence

1. ## Uniform convergence

Hello, I need to investigate Uniform convergence for these two using Weierstrass M-test, 10x in advance.
Assaf.

2. For the first problem note $x^n + x^{-n} \leq 2$.
Thus, $|f_n(x)| \leq \frac{2n^2}{\sqrt{n!}}$ and $\sum_{n=0}^{\infty} \frac{2n^2}{\sqrt{n!}} < \infty$.

I did not do the second problem but I would imagine you need the fact $\log \left( 1 + \frac{x^2}{n\log^2 n} \right) \leq \log \left( 1+\frac{a^2}{n\log^2 n} \right)$.
Now let us hope that $\sum_{n=2}^{\infty} \log \left( 1+\frac{a^2}{n\log^2 n} \right) < \infty$.

3. ## it is

is is < Inf, I used my calculator to see, the question is how I show it...
10x.

4. Originally Posted by asi123
is is < Inf, I used my calculator to see, the question is how I show it...
10x.
It is! And there is an easy way to show it.

You need the fact that $\log (1+x) \leq x$ for $x\geq 0$.
Thus,
$0\leq \log \left( 1 + \frac{a^2}{n^2\log n} \right) \leq \frac{a^2}{n^2\log n}$
And of course, by applying Betrand's test
$\sum_{n=2}^{\infty}\frac{a^2}{n^2\log n} < \infty$

5. Originally Posted by ThePerfectHacker
For the first problem note $x^n + x^{-n} \leq 2$.
$x^n + x^{-n} \leq 2$
How is it true generally?

6. Originally Posted by Isomorphism
$x^n + x^{-n} \leq 2$
How is it true generally?
It should be $x^n + x^{-n}\geq 2$. But that does not solve the problem then.
In that case we should maybe consider $x^n+x^{-n} \leq 2^n+2^{-n}$.