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Ultimate Test for Convergence of infinite series?

  1. Aug 17, 2011 #1

    I'm studying infinite series and am really struggling with memorizing all the tests for convergence in my book, there's like 10 of them. I don't think I'm going to be successful in memorizing all of them. I will never be asked in my course to use a specific test to determine convergence but to determine weather a series converges using a test and showing my work.

    I'm really afraid that on my examinations there are going to be series were the only way I can evaluate them is by using one test only, one that I learned that is, or that I'm going to get my examination and see a question that asks me to determines weather or not a infinite series converges or not, and my mind will just go blank as I go through all the tests that I know in my head, which will be hard to do when I don't really have them all memorized...

    So I was wondering if someone could inform me of the be all end all test for convergence for infinite series, that can be used for any infinite series, or most of them. I wouldn't mind spending some time learning something new, if I can just really get the one technique down and master it. There's got to be some test that can be done on all series, or most of them, that's in some upper math courses that I haven't studied yet.
  2. jcsd
  3. Aug 17, 2011 #2
    I'm very sorry, but I'm afraid such a test does not exist. Some series are better suited for a certain test, while other series are suitable for another test

    Here are some hints though:

    - First, ALWAYS check first if the terms go to 0. This is very easy to do.
    - A test that works in many cases is the integral test. But the integral can be difficult to calculate sometimes
    - if the series involves factorials, ALWAYS try the ratio test. It fails rarely in this cases.
    - if the series involves powers, try the root test. The ratio and root test are interesting in the case of power series
    - Alternating series are easy since they always converge if the terms go to 0
    - If the above all fails or is too difficult, try a comparison test.

    If you keep this list in mind, you should be able to work out most of the series that they throw at you...
  4. Aug 17, 2011 #3
    hmmm thanks
  5. Aug 17, 2011 #4
    I have literally had nightmares about this question until I looked at a good real analysis book.
    Browse through the chapter on infinite series in this book, there are very helpful side comments
    & explanatory remarks about what to do.
  6. Aug 17, 2011 #5
    I take it that the cauchy condensation test is not so common i can't really find any examples and it;s not in my text
  7. Aug 18, 2011 #6


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    The hierarchy I give to my students is:

    1. Test for Divergence (also micromass' #1) -- if the limit of the terms doesn't go to zero, you're done as far as convergence is concerned

    2a. Alternating Series Test (where appropriate) -- alternating series are the easiest to check for convergence, since one condition IS the Test for Divergence and the other is to check that the terms decrease monotonically [when this is too hard to manage algebraically, replacing an with f(x) and showing that df/dx < 0 is often useful]

    If you are asked for conditional vs. absolute convergence, test for divergence first, then test the absolute series for convergence, then finish with the check for monotonic decrease if the series isn't absolutely convergent to see if it's at least conditionally convergent

    2b. The two tests for absolute convergence you'll make the most use of are

    Ratio Test (good for terms with factorials and exponential [ cn ] factors, USELESS for power-law [ np ] factors and rational functions of polynomials)


    Comparison Tests -- the simple Comparison Test and the Limit Comparison Test (good for such things as power-law functions and rational functions of polynomials, but also useful for some other general terms)

    3. The Integral Test is often of help when the Ratio Test and the Comparison Tests aren't, provided you can work out the behavior of the improper integral

    4. Root Test -- you'll find you don't use this one much because it requires the general term of the series to have all of its factors of the form (expression)kn , with k being some real constant, in order to produce a clean test

    Raabe's Test -- I only learned about this one recently; it's an extension of the Ratio Test; a lot of introductory courses don't even cover it.

    There is no ultimate test for series convergence because there are just too many sorts of general terms one could investigate (just as there is no Universal Method for Problem-Solving, alas).
    Last edited: Aug 18, 2011
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