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Divergent infinite series

  1. Sep 30, 2009 #1
    I can't remember much from my intro. analysis class anymore.

    If you have an infinite series that ultimately converges, can the first few terms diverge (i.e., can they move away from the convergence point)? And if so, how many of these terms can do so?

    I'm trying to understand how to "get a feel" for a divergent series. If the first 5000 terms increase, I still can't assume that the rest of the terms will also increase... right?
     
  2. jcsd
  3. Sep 30, 2009 #2
    If the series converges, it doesn't matter what any of the terms are. Take 10n/n! for example and look at its graph

    http://www2.wolframalpha.com/Calculate/MSP/MSP521197dh1fi067g2gh400004fg666f38h6hb56a?MSPStoreType=image/gif&s=21

    It's first few terms are relatively large, yet the sum of the series from 1 to ∞ is e10-1

    The first 5000 terms of 5000n/n! will get very large, but the series from 1 to ∞ will still converge to a number: e5000-1

    You can add as large a number you want, or a finite number of terms to a convergent series and that doesn't affect its convergence, only the sum of the series.
     
    Last edited by a moderator: Apr 24, 2017
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