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Math Help - Two Series & Equality

  1. #1
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    Two Series & Equality

    I have no idea where to start with this one. I tried finding the limits of each, finding the partial sums, I just don't really know how to prove it

    Show that

    \frac{\(-1)^(n+1)}{n} = \frac{\n(2n-1)}{1}

    any help would be appreciated
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  2. #2
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    Quote Originally Posted by starfish View Post
    I have no idea where to start with this one. I tried finding the limits of each, finding the partial sums, I just don't really know how to prove it

    Show that

    \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}=\sum_{n=1}^\infty\frac{1}{2n(2n-1)}

    any help would be appreciated
    Did I guess right in correcting your (messy) formula? (you can click on it to see what I wrote)

    If so, you should cut the left-hand side sum into two parts: the even (the numerator is -1) and the odd terms (it is +1), and then put both sums together.

    Another way (this is the same in fact) is summing the terms two by two: write \sum_{n=1}^\infty a_n=(a_1+a_2)+(a_3+a_4)+\cdots=\sum_{p=1}^\infty( a_{2p-1}+a_{2p}) for the left-hand side.
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