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Infinite Series: A Synopsis

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Useful Inequalities

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Always

 
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Eventually

  1. for any
  2. (!) smile Remember this when using the Test for Divergence
 
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"Eventually"

for any

(!) smile Remember this when using the Test for Divergence

Tips

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Tips for Series

 
  • Often the hardest part of showing convergence or divergence of a series is the indecision: What do I believe it does? After all, you'll have a tough time showing a series converges if it doesn't!
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  • The limits listed in another section can help a lot with the Test for Divergence. Together with inequalities you can often get an idea of what to try to show. If the individual terms of the series "look like" as then the series "looks like" 1/n and you will want to show it diverges, perhaps even setting up a comparison, or limit comparison with 1/n itself.
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(see the transcript file for additional information)
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  • The limits listed in another section can help a lot with the Test for Divergence. Together with inequalities you can often get an idea of what to try to show. If the individual terms of the series "look like" as then the series "looks like" and you will want to show it diverges, perhaps even setting up a comparison, or limit comparison with 1/n itself.
  • Many limits boil down to "look like" ratios of polynomials after stripping out trig functions using the Useful Inequalities for trig functions.
  • The eventual behavior that for any leads to the peculiar rule of thumb that in lots of ratios ln(n) "looks like" 1 since any positive power of n will dominate it:
    • informally, "looks like" so converges
    • more carefully, (eventually),
      so,
      which is a convergent p-series.
 
-- DickFurnas - 21 Oct 2008
 
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