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Math Help - General question on proving the convergence of a series

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    General question on proving the convergence of a series

    Sorry no specific example here but I was just wondering if it's mathematically correct to prove that a series is convergent by comparing it to another series that is larger than it and use an integral test on that larger series to prove that it is convergent.

    I ask this because all the examples I have seen of the "comparison test" try to compare that series to a series in the "p-series form" where you can tell straight away if it's convergent by looking at the power of the bottom variable. The problem is that I have trouble finding proper comparisons because I get all confused. So is it wrong if I use both a comparison and an integral test in unison?
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    Re: General question on proving the convergence of a series

    Hey jgv115.

    You can do this but you should outline the type of series. For example is an alternating series or are all terms the same sign?
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    Re: General question on proving the convergence of a series

    Quote Originally Posted by jgv115 View Post
    Sorry no specific example here but I was just wondering if it's mathematically correct to prove that a series is convergent by comparing it to another series that is larger than it and use an integral test on that larger series to prove that it is convergent.

    I ask this because all the examples I have seen of the "comparison test" try to compare that series to a series in the "p-series form" where you can tell straight away if it's convergent by looking at the power of the bottom variable. The problem is that I have trouble finding proper comparisons because I get all confused. So is it wrong if I use both a comparison and an integral test in unison?
    You need to understand that in a positive-term series, if \displaystyle \begin{align*} a_n \leq b_n \end{align*} for all n, then \displaystyle \begin{align*} \sum {a_n} \leq \sum b_n \end{align*}.

    Now if \displaystyle \begin{align*} \sum b_n \end{align*} is convergent, that means it can be equated to a number, and so anything less than or equal to it is also a number. Therefore \displaystyle \begin{align*} \sum a_n \end{align*} will also be convergent.

    Meanwhile, if \displaystyle \begin{align*} \sum a_n \end{align*} is divergent, it means that the series will shoot off to \displaystyle \begin{align*} \infty \end{align*}. Anything greater than or equal to this will also shoot of to \displaystyle \begin{align*} \infty \end{align*} (quicker), and so \displaystyle \begin{align*} \sum b_n \end{align*} is also divergent.
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    Re: General question on proving the convergence of a series

    Hi chiro,

    Yes they are all the same sign. I am aware of the tests required for an alternating series!

    Thanks for your confirmation.

    edit: Yep, I understand, Prove it!
    Last edited by jgv115; June 8th 2013 at 12:42 AM.
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    Re: General question on proving the convergence of a series

    Remember than when using the first comparison test which prove it mention both series must be the sum of positive terms.
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    Re: General question on proving the convergence of a series

    Quote Originally Posted by Shakarri View Post
    Remember than when using the first comparison test which prove it mention both series must be the sum of positive terms.
    And if your series is not all positive term, you can test the series of absolute values against another series. If a series is absolutely convergent, then it is still convergent.

    Proof:

    \displaystyle \begin{align*} - \left| a_n \right| \leq a_n &\leq \left| a_n \right| \\ \sum { -\left| a_n \right| } \leq \sum a_n &\leq \sum \left| a_n \right| \end{align*}

    and so by the comparison test, if \displaystyle \begin{align*} \sum \left| a_n \right| \end{align*} is convergent, therefore so must be \displaystyle \begin{align*} \sum a_n \end{align*}.
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