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Thread: Identity

  1. #1
    MHF Contributor red_dog's Avatar
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    Identity

    There is another proof, without induction, for this identity?

    $\displaystyle 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\ldots+\frac{1}{2n-1}-\frac{1}{2n}=\frac{1}{n+1}+\frac{1}{n+2}+\ldots+\f rac{1}{2n}$
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    This is Catalan's identity.

    Consider $\displaystyle k_n = 1 + \frac{1}
    {2} + \cdots + \frac{1}
    {n}.$ Note that $\displaystyle \frac{1}
    {2}k_n = \frac{1}
    {2} + \frac{1}
    {4} + \cdots + \frac{1}
    {{2n}}.$ The LHS is $\displaystyle k_{2n} - 2\left( {\frac{1}
    {2}k_n } \right)$ & the RHS is $\displaystyle k_{2n}-k_n,$ as required $\displaystyle \blacksquare$
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