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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?
    Thanks.


    2. Relevant equations



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

    CompuChip

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    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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