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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

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.