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Can two different infinite series converge to the same limit?

  1. Dec 27, 2011 #1

    The title of the thread doesn't adequately describe the question I want to ask, so here it is:

    Suppose we have two infinite series, [itex]\sum_{n=1}^{\infty}a_n[/itex] and [itex]\sum_{n=1}^{\infty}b_n[/itex], both of which are convergent. Also suppose [itex]a_n \leq b_n[/itex] for all [itex]n[/itex], and [itex]a_n < b_n[/itex] for at least one [itex]n[/itex]. Is it possible that they both converge to the same limit? Or can we say that [itex]\sum_{n=1}^{\infty}a_n[/itex] is strictly smaller than [itex]\sum_{n=1}^{\infty}b_n[/itex]?

    If the answer to the above question is that they can have the same limit, then what if [itex]a_n<b_n[/itex] for infinitely many [itex]n[/itex]?

  2. jcsd
  3. Dec 27, 2011 #2


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    Gold Member

    Suppose [itex]a_i < b_i[/itex] and that [itex]a_n \leq b_n[/itex] for all [itex]n \neq i[/itex]. Then if [itex]\sum_{n \in \mathbb{N}}a_n[/itex] and [itex]\sum_{n \in \mathbb{N}}b_n[/itex] converge, it is easy to show the following:
    [tex]\sum_{n \leq i}a_n < \sum_{n \leq i}b_n[/tex]
    [tex]\sum_{n > i}a_n \leq \sum_{n > i}b_n[/tex]
    Adding these together gives
    [tex]\sum_{n \in \mathbb{N}}a_n = \sum_{n \leq i}a_n + \sum_{n > i}a_n < \sum_{n \leq i}b_n + \sum_{n > i}b_n = \sum_{n \in \mathbb{N}}b_n[/tex]
    This should answer your question.
  4. Dec 27, 2011 #3


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    Science Advisor

    There is no way they can have the same limit. If you compare finite sums including at least one n where an < bn, then the a partial sum will be smaller and there is no way it can make up the difference without a term with a > b.
  5. Dec 27, 2011 #4
    Ok, that's great, thanks!
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