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Math Help - Convergence of alternating series

  1. #1
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    Convergence of alternating series

    Let a_n\in\mathbb{R} and \lim_{n\to\infty}a_n=0.
    Prove or provide a counter example that \sum_{n=1}^{\infty}(-1)^{n+1}a_n^2 converges.

    I was wondering wether this is a valid counter example:

    Let a_n=\begin{cases}\frac{1}{\sqrt[4]{n}}& n \text{ odd }\\\frac{1}{\sqrt{n}}& n\text{ even }\end{cases}.

    Then

    a_n^2=\begin{cases}\frac{1}{\sqrt{n}}& n \text{ odd }\\ \frac{1}{n}& n\text{ even }\end{cases}

    And \sum_{n=1}^{\infty}(-1)^{n+1}a_n^2=\sum_{n=1}^{\infty}\left(\frac{1}{\s  qrt{2n+1}}-\frac{1}{2n}\right).

    Now using the integral test we find that this series converges if and only if

    \int_1^{\infty}\left(\frac{1}{\sqrt{2x+1}}-\frac{1}{2x}\right)dx converges.

    Now:

    \int_1^{\infty}\left(\frac{1}{\sqrt{2x+1}}-\frac{1}{2x}\right)dx=\left[\sqrt{2x+1}-\frac{1}{2}ln(2x)\right]_1^{\infty}=\left[\sqrt{2x+1}-ln(\sqrt{2x})\right]_1^{\infty}=\infty

    So that the series diverges.

    Is this correct, and if so is there a more straight forward way to do this?

    Thank you.
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  2. #2
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    Good example. You could have made it slightly simpler by taking a_n=\begin{cases}0& n \text{ odd,}\\ \frac{1}{\sqrt n}& n\text{ even.}\end{cases}
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  3. #3
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    Thank you Opalg.

    Now I am surprised I didn't figure out the example you gave.
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