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Limits of Series

  1. Oct 23, 2004 #1
    How do you prove the limit of a series?

    For example, bn=1/square root of (n^2+1)+ 1/square root of (n^2+2) + ...+1/square root of (n^2 + n). I know this series converges since the summation of 1/n^2 does. But the only reason this is because of integration. How could I know that some series converges without using integrals, and then be able to find to what it converges?
  2. jcsd
  3. Oct 24, 2004 #2
    Do you mean the series:

    [tex]\lim_{n \to \infty} \sum_{i=1}^n \frac{1}{\sqrt{n^2 + n}} \ \ ?[/tex]

    Because that series does not converge.
  4. Oct 24, 2004 #3
    Doesn't it converge to 1?
  5. Oct 24, 2004 #4

    matt grime

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    Nope, since

    [tex]n^2+n < n^2+2n+1[/tex]

    the sum you've got there, modulo some initial terms, is greater than

    [tex] \sum \frac{1}{n+1}[/tex]

    so it diverges.

    NB. cogito's post uses n twice as the parameter and the limit, it should be i inside the sum.
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