# Convergence in mean square (or L^2) sense

1. Nov 9, 2009

### kingwinner

Convergence in "mean square" (or L^2) sense

1. The problem statement, all variables and given/known data
This is an example from a textbook (with solutions) in which I am feeling confused.
Let fn(x) = [n/(1+n2x2)] - (n-1)/[1+(n-1)2x2] in the interval 0<x<L. This series telescopes so that
N
∑ fn(x) = N/(1+N2x2)
n=1

L
∫ [∑ fn(x)]2 dx =
0
L
∫ N2/(1+N2x2)2 dx =
0
NL
∫ N/(1+y2)2 dy (let y=Nx)
0
This last line -> +∞ as N->∞
Since it does not converge to 0, the series does NOT converge in the mean-square (or L2) sense to f(x)=0.

2. Relevant equations/concepts
Convergence in mean square/L2 sense

3. The attempt at a solution
N/A

(i) Now I don't understand why we have to use the change of variable y=Nx. What is the point of doing this?

(ii) Also, WHY as N->∞,
NL
∫ N/(1+y2)2 dy -> +∞ ?
0