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

Assaf.

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- Jun 2nd 2008, 05:09 AMasi123Uniform convergence
Hello, I need to investigate Uniform convergence for these two using Weierstrass M-test, 10x in advance.

Assaf. - Jun 2nd 2008, 04:39 PMThePerfectHacker
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 $\displaystyle \sum_{n=2}^{\infty} \log \left( 1+\frac{a^2}{n\log^2 n} \right) < \infty$. - Jun 2nd 2008, 08:50 PMasi123it is
is is < Inf, I used my calculator to see, the question is how I show it...

10x. - Jun 2nd 2008, 08:57 PMThePerfectHacker
It is! And there is an easy way to show it. (Happy)

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

$\displaystyle \sum_{n=2}^{\infty}\frac{a^2}{n^2\log n} < \infty$ - Jun 3rd 2008, 07:50 AMIsomorphism
- Jun 3rd 2008, 07:55 AMThePerfectHacker