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Homework Help: Testing series for convergence

  1. Mar 3, 2009 #1
    1. The problem statement, all variables and given/known data
    On a recent homework problem, I tested a series for convergence using the comparison test. If the first term of the series to be tested, a_n, was included, my test was inconclusive. From a problem that the lecturer did in class, he stated that the first few terms of an inifinte series don't matter when testing for convergence. So I compared the sum of the series from n=2 to infinity to my chosen series, showed it was smaller than this convergent series and this implied convergence of a_n.
    The TA had my comment marked incorrect, stating that the first few terms can't be disregarded, so I'm unsure which is true.
    Is it a case where, if you're calculating the sum, you can't disregard the first few terms, but if you're only testing for convergence, you can?

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Mar 3, 2009 #2


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    Homework Helper

    Let us assume that all the terms in the sum are finite (e.g. no [itex]a_n[/itex] is infinite). Then any finite sum
    [tex]\sum_{n = 1}^k a_n[/tex]
    is finite. Since
    [tex]\sum_{n = 1}^\infty a_n = \sum_{n = 1}^k a_n + \sum_{n = k + 1}^\infty a_n[/tex]
    there are two possibilities: the sum from k + 1 to infinity is finite, in which case the expression is (something finite + something finite = something finite), or it is infinite, and then it is (something finite + something infinite = infinite).
    Of course, for calculating the value of the sum, you cannot neglect any terms.

    There can be subtleties with non-absolutely converging sums and such, if you have any doubt I think the best thing to do is ask your TA why he marked it incorrect (e.g. if he can give you an example where it goes wrong) and/or go see the teacher.
  4. Mar 4, 2009 #3
    Thanks Compuchip, I appreciate the reply.
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