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 Can someone please explain? Thank you!