Convergence (series)

**fqqs** · May 26th 2012, 09:06 AM

I have to assess if $\sum_{n=1}^{ \infty } \frac{1}{1+ n^{2} \cdot x^{2} }$ is convergent (uniformly convergent)

please help

**Plato** · May 26th 2012, 10:08 AM

Originally Posted by fqqs

I have to assess if $\sum_{n=1}^{ \infty } \frac{1}{1+ n^{2} \cdot x^{2} }$ is convergent (uniformly convergent)

Think of the limit comparison test.

What can you say about $\left( {\frac{{{n^2}}}{{1 + {n^2}{x^2}}}} \right) \to~?$

**fqqs** · May 26th 2012, 10:20 AM

Originally Posted by Plato

Think of the limit comparison test.

What can you say about $\left( {\frac{{{n^2}}}{{1 + {n^2}{x^2}}}} \right) \to~?$

It goes to $\frac{1}{x^{2}}$ ?

**Plato** · May 26th 2012, 10:48 AM

Originally Posted by fqqs

It goes to $\frac{1}{x^{2}}$ ?

What does that tell you about the values of $x$ for which it converges or diverges?

**fqqs** · May 26th 2012, 11:05 AM

it converges for every $x \ne 0$

?

**Plato** · May 26th 2012, 11:48 AM

Originally Posted by fqqs

it converges for every $x \ne 0$

Correct

**fqqs** · May 26th 2012, 12:18 PM

but that only means that it is pointwisely convergent for every $x \ne 0$ , yes?

so i need to check if it is pointwisely convergent somewhere , yes?

maybe I should use cases when $|x| < 1$ and when $|x| \ge 1$ and use Weierstrass?

**fqqs** · June 1st 2012, 10:33 AM

upppp

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