I have a problem about an infinite sum

Quick description

This page attempts to direct you to appropriate advice for dealing with your infinite sum by asking a few questions to narrow down what your problem is. If you click on the answers, you will get more text and/or further questions.

Prerequisites

Basic real analysis

What is your problem?

Which of the following statements best describes your problem?

I want to calculate an infinite sum

In order for you to be able to calculate an infinite sum, it has to be fairly special. A common kind of specialness is that there is another way of thinking about the infinite sum. And the most common alternative way of thinking about the infinite sum is to relate it to some kind of series expansion of some function. More details can be found in the article If you want to calculate an infinite sum exactly, relate it to coefficients associated with some function. Sometimes you can evaluate an infinite sum by reexpressing it as a telescoping sum: see to calculate an infinite sum exactly, try antidifferencing.

I have an explicitly defined infinite sum and I want to bound it or prove that it converges

Do you believe your sum to be absolutely convergent?
- Yes
  - Then the most useful single principle to bear in mind is the comparison test: roughly speaking, if you can find a larger series that converges, then so does yours. Several other tests for convergence can be derived from the comparison test. See how to bound a sum of infinitely many positive real numbers for more details.
- No
  - Then a technique that often helps is the alternating series test: if you have a sequence $(a_n)$ that is decreasing and tends to zero, then the sum $\sum_{n=1}^\infty(-1)^na_n$ converges. This is a special case of Abel's test, which can be proved using partial summation and is discussed in that article. For more on proving conditional convergence, see how to prove conditional convergence.

I have an infinite sum and I want to prove that it converges. I do not know what the terms in the sum are, but I do know certain properties that they have.

What to do here depends very much on what those properties are. An obviously helpful property is that the terms $a_n$ in your infinite sum are bounded by a sequence of terms $b_n$ that you already know to converge. In that case, you can simply use the comparison test. If you have an infinite sum of the form $\sum_na_nb_n$ , then it may be that you have information about the rates at the sequences $(a_n)$ and $(b_n)$ shrink or grow as $n$ tends to infinity. Then you may need to use inequalities such as Hölder's inequality.