I need help with the following problem.(adsbygoogle = window.adsbygoogle || []).push({});

Consider the serie of function

[tex]\sum_{n=1}^{\infty}\frac{1}{1+n^2x}[/tex]

The serie is undefined for [itex]x \in \{0\}\cup \{-1/n^2, \ n\in \mathbb{N}\}[/itex]. I want to find wheter it converges pointwise in (-1, 0) or not and if it does, does it converge uniformly?

The way I would start this problem is by saying: For a given number [itex]m \in \mathbb{N}[/itex], consider

[tex]x_0 \in \left(\frac{-1}{m^2} \ ,\frac{-1}{(m+1)^2}\right)[/tex]

Consider

[tex]f_n(x) = \frac{1}{1+n^2x}[/tex]

Then

[tex]|f_n(x_0)| = \frac{1}{|1+n^2x_0|} = \frac{1}{|1-n^2|x_0||}= \left\{ \begin{array}{rcl}

\frac{1}{1-n^2|x_0|} & \mbox{for}

& n<\sqrt{\frac{1}{|x_0|} \\

\frac{1}{n^2|x_0|-1} & \mbox{for}

& n>\sqrt{\frac{1}{|x_0|}

\end{array}\right [/tex]

and

[tex]\sum_{n=1}^{\infty}|f_n(x_0)| = \sum_{n=1}^{\left[\sqrt{1/|x_0|}\right]}\frac{1}{1-n^2|x_0|} + \sum_{n=\left[\sqrt{1/|x_0|}\right]+1}^{\infty}\frac{1}{n^2|x_0|-1}[/tex]

I'm guessing this serie converges, but I'm having trouble finding a convergent serie to bound it with. The other convergence tests have failed and the use of the integral convergence criterion is forbiden. I know that if there is a serie to bound it with, it would be of the form

[tex]\sum_{n=1}^{\infty}a_n = \sum_{n=1}^{\left[\sqrt{1/|x_0|}\right]}\frac{1}{1-n^2|x_0|} + \sum_{n=\left[\sqrt{1/|x_0|}\right]+1}^{\infty}b_n[/tex]

with

[tex]\frac{1}{n^2|x_0|-1} \leq b_n[/tex]

for n > N.

Edit:

And if there exists such an N that also satisfies

[tex]N\leq \left[\sqrt{1/|x_0|}\right][/tex]

then according to Weirstrass M-test, the convergence is uniform.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Convergence of serie

**Physics Forums | Science Articles, Homework Help, Discussion**