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Math Help - Infinite Series Convergence

  1. #1
    Senior Member MaxJasper's Avatar
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    Question Infinite Series Convergence

    Hi folks,

    Please comment on how to proceed with finding whether the following series converge or not:
    1+\frac{1}{2}+\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}-\frac{1}{10}\text{--}\text{++}+\text{--}-

    Thanks a lot.

    Max.
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  2. #2
    MHF Contributor

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    Re: Infinite Series Convergence

    Quote Originally Posted by MaxJasper View Post
    Please comment on how to proceed with finding whether the following series converge or not:
    1+\frac{1}{2}+\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}-\frac{1}{10}\text{--}\text{++}+\text{--}-
    This is only a comment or suggestion. Let
     \begin{align*}s_1&=\frac{1}{1}+\frac{1}{2}+\frac{1  }{3} \\ s_2&=\frac{1}{4}+\frac{1}{5}+\frac{1}{6} \\&\vdots \\ s_n&=\frac{1}{3n-2}+\frac{1}{3n-1}+\frac{1}{3n}\end{align*}

    Now your series is \sum\limits_{n = 1}^\infty  {{{\left( { - 1} \right)}^{n + 1}}{s_n}}.
    Does the alternating series test apply?
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  3. #3
    Senior Member MaxJasper's Avatar
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    Thumbs up Re: Infinite Series Convergence

    What a beauty! Thanks a lot bro...very simple and nice.
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  4. #4
    GJA
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    Re: Infinite Series Convergence

    Had a response, but Plato's solution is very nice.
    Last edited by GJA; August 16th 2012 at 03:09 PM.
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