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 $|f_k (x)| \le g_k (x)$ 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 $x$ $$ \left| \sum_{k=0}^n f_k(x) \right |\le \sum_{k=0}^n |f_k(x)| \le \sum_{k=0}^n g_k(x) \le \sum_{k=0}^\infty g_k(x),$$ 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 $$\pi \sum_{k=1}^\infty {e^{-k} \over k^2}$$