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Math Help - Sumatory

  1. #1
    Member Nacho's Avatar
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    Sumatory

    I canīt solve this sumatory

    <br />
\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c}<br />
   n  \\<br />
   k  \\<br /> <br />
 \end{array} } \right)\frac{{\left( { - 1} \right)^k }}<br />
{{2k + 1}}} <br />

    I have my doubt if exist... thanks!
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    \sum\limits_{k=0}^{n}{\binom nk\frac{(-1)^{k}}{2k+1}}=\int_{0}^{1}{\left\{ \sum\limits_{k=0}^{n}{\binom nk\left( -x^{2} \right)^{k}} \right\}}=\int_{0}^{1}{\left( 1+x^{2} \right)^{n}\,dx}.
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  3. #3
    Member Nacho's Avatar
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    Quote Originally Posted by Krizalid View Post
    \sum\limits_{k=0}^{n}{\binom nk\frac{(-1)^{k}}{2k+1}}=\int_{0}^{1}{\left\{ \sum\limits_{k=0}^{n}{\binom nk\left( -x^{2} \right)^{k}} \right\}}=\int_{0}^{1}{\left( 1+x^{2} \right)^{n}\,dx}.
    In fact, this sumatory born of this integral then my doubt is solve this sumatory

    Thanks Kriza
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