1. ## Convergence (series)

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

2. ## Re: Convergence (series)

Originally Posted by fqqs
I have to assess if $\displaystyle \sum_{n=1}^{ \infty } \frac{1}{1+ n^{2} \cdot x^{2} }$ is convergent (uniformly convergent)
Think of the limit comparison test.

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

3. ## Re: Convergence (series)

Originally Posted by Plato
Think of the limit comparison test.

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

4. ## Re: Convergence (series)

Originally Posted by fqqs
It goes to $\displaystyle \frac{1}{x^{2}}$?
What does that tell you about the values of $\displaystyle x$ for which it converges or diverges?

5. ## Re: Convergence (series)

it converges for every $\displaystyle x \ne 0$

?

6. ## Re: Convergence (series)

Originally Posted by fqqs
it converges for every $\displaystyle x \ne 0$
Correct

7. ## Re: Convergence (series)

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

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

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

upppp