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Math Help - [SOLVED] Application of Binomial theorem?

  1. #1
    Super Member fardeen_gen's Avatar
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    [SOLVED] Application of Binomial theorem?

    Prove that:
    \sum\limits_{r = 0}^{n} {{n\choose r}\frac{(\cos 2x)^{r + 1}}{r + 1}} = \frac{2^{n + 1}{\cos}^{2(n + 1)}x - 1}{n + 1}
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by fardeen_gen View Post
    Prove that:
    \sum\limits_{r = 0}^{n} {{n\choose r}\frac{(\cos 2x)^{r + 1}}{r + 1}} = \frac{2^{n + 1}{\cos}^{2(n + 1)}x - 1}{n + 1}
    2\cos^2(x)-1=\cos(2x)

    so:

    2^{n+1}\cos^{2(n+1)}(x)=(\cos(2x)+1)^{n+1}

    Use the binomial expansion of the right hand side above to show that:


    (\cos(2x)+1)^{n+1}=1+(n+1)\left[\sum\limits_{r = 0}^{n} {{n\choose r}\frac{(\cos 2x)^{r + 1}}{r + 1}} \right]

    CB
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