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Math Help - Testing for Convergence of Series

  1. #1
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    Unhappy Testing for Convergence of Series

    Hello, and thanks for help in advance. I have an exam tomorrow that I'm very nervous about, so here goes:

    I understand the justification for quite a few of the tests for convergence or divergence of series, such as the Comparison & Integral tests, but often have trouble applying any of them.

    One example is the following:

    If from n = 1 to n = infinity, is 1/(n + 4)^(1/2) convergent or divergent?

    I understand that for p-series 1 / n^p, the function converges where p > 1 and diverges otherwise. Can I then say, because

    1/(n + 4)^(1/2) ~ 1/(n)^(1/2) where 1/(n)^(1/2) diverges

    that my original series (on left) diverges?

    Are approximation methods like this (whether or not this one specifically works) allowed?
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  2. #2
    Moo
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    Hello,

    You can, but you can also do it this way :

    change the indice :

    \sum_{n=1}^\infty \frac{1}{(n+4)^{1/2}}=\sum_{n=5}^\infty \frac{1}{n^{1/2}}<\sum_{n=1}^\infty \frac{1}{n^{1/2}}
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  3. #3
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    That is a good way of justifying the ~.

    Are there any examples that make simplification via a change of the indice more understandable?
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