Tính tổng $S = \left ( \frac{C_{2013}^0 }{1} \right )^{2}+\left ( \frac{C_{2013}^1 }{2} \right )^{2}+...+\left ( \frac{C_{2013}^{2013}}{2014} \right )^{2}$
$S = \left ( \frac{C_{2013}^0 }{1} \right )^{2}+...+\left ( \frac{C_{2013}^{2013}}{2014} \right )^{2}$
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Bắt đầu bởi Thuat ngu, 12-02-2017 - 15:38
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Đã gửi 23-03-2017 - 00:45
hxthanh
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Tín đồ $\sum$
$S=\sum_{k=0}^{2013}\dfrac{\binom{2013}{k}^2}{(k+1)^2}$
$S=\dfrac{1}{2014^2}\sum_{k=0}^{2013}\binom{2014}{k+1}^2$
$S=-\dfrac{1}{2014^2}+\dfrac{1}{2014^2}\sum_{k=0}^{2014}\binom{2014}{k}^2$
$S=-\dfrac{1}{2014^2}+\dfrac{1}{2014^2}\sum_{k=0}^{2014}\binom{2014}{k}\binom{2014}{2014-k}$
$S=-\dfrac{1}{2014^2}+\dfrac{1}{2014^2}\binom{4028}{2014}\qquad$ (Vandermonde)
$S=\dfrac{1}{2014^2}\sum_{k=0}^{2013}\binom{2014}{k+1}^2$
$S=-\dfrac{1}{2014^2}+\dfrac{1}{2014^2}\sum_{k=0}^{2014}\binom{2014}{k}^2$
$S=-\dfrac{1}{2014^2}+\dfrac{1}{2014^2}\sum_{k=0}^{2014}\binom{2014}{k}\binom{2014}{2014-k}$
$S=-\dfrac{1}{2014^2}+\dfrac{1}{2014^2}\binom{4028}{2014}\qquad$ (Vandermonde)
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