Convergence in mean square (or L^2) sense

[kingwinner]

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

Homework Statement

This is an example from a textbook (with solutions) in which I am feeling confused.
Let f_n(x) = [n/(1+n²x²)] - (n-1)/[1+(n-1)²x²] in the interval 0<x<L. This series telescopes so that
N
∑ f_n(x) = N/(1+N²x²)
n=1

L
∫ [∑ f_n(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. Homework Equations /concepts
Convergence in mean square/L² sense

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

Can someone please explain?
Thank you!

 

[kingwinner]

Why is it a good idea to bring N into the limits of integration? After the change of variable, N appears both in the upper limit of integration AND in the integrand...