Tricki

## How to bound a sum of infinitely many positive real numbers

### Quick description

This article is about standard tests for convergence of sums of positive real numbers, and how to use them. The most obvious is the comparison test, but others that are sometimes useful are the integral test and the condensation test.

 Incomplete This article is incomplete. More examples needed. So far there are no examples for the integral or condensation tests. It would be nice to do the same series by these two methods.

### Example 1

Here is a well-known proof that the series converges. First, the partial sums form an increasing sequence. Secondly, for each , (since . And third, . Therefore, the partial sums form an increasing sequence that is bounded above, which implies that the series converges.

### General discussion

The technique we have just used is called the comparison test. The precise statement of the test is as follows.

Theorem 1 Let and be two sequences of non-negative real numbers, and suppose that for every . Then if the series converges, so does the series .

The proof of the theorem can be abstracted out of the argument in Example 1. For every , we have , so the partial sums of the series are increasing and bounded above.

It is often useful to have a more flexible version of the comparison test, which follows easily from the version as stated above. Here it is.

Corollary 2 Let and be two sequences of non-negative real numbers, and suppose that there exist and such that for every . Then if the series converges, so does the series .

To deduce this from Theorem 1, observe first that if the series converges, then so does the series . But changing finitely many terms of an infinite series does not affect whether or not it converges, so the series converges as well. Therefore, by Theorem 1, so does the series .

### Example 2

As an illustration of how to use the slightly more general comparison test, consider the infinite sum . A slightly cheeky, but perfectly rigorous, argument is to say that if , then , and , so . Since we have established that converges, so does .