- #1

twoflower

- 368

- 0

I have few questions about this excercise:

**Analyse pointwise, uniform and local uniform convergence of this series of functions:**

[tex]

\sum_{k=2}^{\infty}\log \left(1 + \frac{x^2}{k \log^2 k} \right)

[/tex]

I'm trying to do it using Weierstrass' criterion. To recall it, it says

[tex]

\mbox{Let } f_n \mbox{ are defined on } 0 \neq M \subset \mathbb{R}\mbox{, let }

S_n := \sup_{x \in M} \left| f_{n}(x)\right|, n \in \mathbb{N}. \mbox{ If }

\sum_{n=1}^{\infty} S_n < \infty\mbox{, then } \sum_{n=1}^{\infty} f_{n}(x) \rightrightarrows \mbox{ on } M.

[/tex]

How to find

[tex]

\sup_{x \in M} \left| f_{n}(x)\right|

[/tex]

?

The derivative is

[tex]

\left(\log \left(1 + \frac{x^2}{k \log^2 k} \right)\right)^{'} = \frac{2x}{k\log^2 k + x^2}

[/tex]

It means that the function is growing for [itex]x > 0[/itex].

**x**going to infinity would bring us problems, so I will take [itex]x \in [-K, K][/itex], where [itex]-\infty < -K < K < \infty[/itex].

Then

[tex]

\sup_{x \in M} \left| f_{n}(x)\right| = \log \left( 1 + \frac{K^2}{n \log^{2} n}\right)

[/tex]

But I don't know how to prove that

[tex]

\sum_{n=2}^{\infty} \log \left( 1 + \frac{K^2}{n \log^{2} n}\right) \mbox{ converges}

[/tex]

Could someone point me to the right direction please?

Thank you.