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Math Help - Identity

  1. #1
    MHF Contributor red_dog's Avatar
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    Identity

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

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