# 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^{} ${f}_{k}:M\to \mathbb{R}$ (The index $k$ runs over
non-negative integers.) on some measure space^{} $M$
and can find another sequence of measurable
functions ${g}_{k}:M\to \mathbb{R}$ such that $|{f}_{k}(x)|\le {g}_{k}(x)$ for all $k$ and almost all $x$ and ${\sum}_{k=0}^{\mathrm{\infty}}{g}_{k}(x)$
converges for almost all $x\in M$ and $$. Then

$${\int}_{M}\sum _{k=0}^{\mathrm{\infty}}{f}_{k}(x)dx=\sum _{k=0}^{\mathrm{\infty}}{\int}_{M}{f}_{k}(x)\mathit{d}x$$ |

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^{}, $$.
Since
${\sum}_{k=0}^{\mathrm{\infty}}{g}_{k}(x)$ converges for almost all $x$,

$$\left|\sum _{k=0}^{n}{f}_{k}(x)\right|\le \sum _{k=0}^{n}|{f}_{k}(x)|\le \sum _{k=0}^{n}{g}_{k}(x)\le \sum _{k=0}^{\mathrm{\infty}}{g}_{k}(x),$$ |

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:

$${\int}_{-\mathrm{\infty}}^{+\mathrm{\infty}}\sum _{k=1}^{\mathrm{\infty}}\frac{\mathrm{cos}(x/k)}{{x}^{2}+{k}^{4}}dx$$ |

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 ${\int}_{-\mathrm{\infty}}^{+\mathrm{\infty}}{g}_{k}(x)\mathit{d}x=\pi /{k}^{2}$ and, as $$, we can interchange summation and integration:

$$\sum _{k=1}^{\mathrm{\infty}}{\int}_{-\mathrm{\infty}}^{+\mathrm{\infty}}\frac{\mathrm{cos}(x/k)}{{x}^{2}+{k}^{4}}\mathit{d}x.$$ |

Doing the integrals, we obtain the answer

$$\pi \sum _{k=1}^{\mathrm{\infty}}\frac{{e}^{-k}}{{k}^{2}}$$ |

Title | criterion for interchanging summation and integration |
---|---|

Canonical name | CriterionForInterchangingSummationAndIntegration |

Date of creation | 2013-03-22 16:20:05 |

Last modified on | 2013-03-22 16:20:05 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 9 |

Author | rspuzio (6075) |

Entry type | Result |

Classification | msc 28A20 |