# criterion for interchanging summation and integration

The following criterion for interchanging integration and summation is often useful in practise: Suppose one has a sequence of measurable functions  $f_{k}\colon M\to\mathbb{R}$ (The index $k$ runs over non-negative integers.) on some measure space  $M$ and can find another sequence of measurable functions $g_{k}\colon M\to\mathbb{R}$ such that $|f_{k}(x)|\leq g_{k}(x)$ for all $k$ and almost all $x$ and $\sum_{k=0}^{\infty}g_{k}(x)$ converges for almost all $x\in M$ and $\sum_{k=0}^{\infty}\int g_{k}(x)\,dx<\infty$. Then

 $\int_{M}\sum_{k=0}^{\infty}f_{k}(x)\,dx=\sum_{k=0}^{\infty}\int_{M}f_{k}(x)\,dx$

This criterion is a corollary of the monotone  and dominated convergence theorems. Since the $g_{k}$’s are nonnegative, the sequence of partial sums is increasing, hence, by the monotone convergence theorem  , $\int_{M}\sum_{k=0}^{\infty}g_{k}(x)\,dx<\infty$. Since $\sum_{k=0}^{\infty}g_{k}(x)$ converges for almost all $x$,

 $\left|\sum_{k=0}^{n}f_{k}(x)\right|\leq\sum_{k=0}^{n}|f_{k}(x)|\leq\sum_{k=0}^% {n}g_{k}(x)\leq\sum_{k=0}^{\infty}g_{k}(x),$

the dominated convergence theorem implies that we may integrate the sequence of partial sums term-by-term, which is tantamount to saying that we may switch integration and summation.

As an example of this method, consider the following:

 $\int_{-\infty}^{+\infty}\sum_{k=1}^{\infty}{\cos(x/k)\over x^{2}+k^{4}}\,dx$

The idea behind the method is to pick our $g$’s as simple as possible so that it is easy to integrate them and apply the criterion. A good choice here is $g_{k}(x)=1/(x^{2}+k^{4})$. We then have $\int_{-\infty}^{+\infty}g_{k}(x)\,dx=\pi/k^{2}$ and, as $\sum_{k=1}^{\infty}k^{-2}<\infty$, we can interchange summation and integration:

 $\sum_{k=1}^{\infty}\int_{-\infty}^{+\infty}{\cos(x/k)\over x^{2}+k^{4}}\,dx.$

Doing the integrals, we obtain the answer

 $\pi\sum_{k=1}^{\infty}{e^{-k}\over k^{2}}$
Title criterion for interchanging summation and integration CriterionForInterchangingSummationAndIntegration 2013-03-22 16:20:05 2013-03-22 16:20:05 rspuzio (6075) rspuzio (6075) 9 rspuzio (6075) Result msc 28A20