sum_{n=4}^{infty} frac{6}{n^{2}-1}

Question

Accepted Answer

Step:1 ：Let (S = sum_{n=4}^{infty} frac{6}{n^{2}-1} = frac{6}{15} + frac{6}{23} + frac{6}{35} + frac{6}{49} + dotsb).Step:2 ：Then (n^{2}S = sum_{n=4}^{infty} frac{6n^{2}}{n^{2}-1} = 6 + frac{6}{2} + frac{6}{3} + frac{6}{4} + dotsb).Step:3 ：Subtracting the first equation from the second gives us (S = 6 + frac{3}{2} + frac{2}{3} + frac{3}{4} + dots = 6 + frac{frac{3}{2}}{1-frac{1}{2}} = 6 + 3 = boxed{9}).

Answer

Step: 1 ：Let (S = sum_{n=4}^{infty} frac{6}{n^{2}-1} = frac{6}{15} + frac{6}{23} + frac{6}{35} + frac{6}{49} + dotsb).Step: 2 ：Then (n^{2}S = sum_{n=4}^{infty} frac{6n^{2}}{n^{2}-1} = 6 + frac{6}{2} + frac{6}{3} + frac{6}{4} + dotsb).Step: 3 ：Subtracting the first equation from the second gives us (S = 6 + frac{3}{2} + frac{2}{3} + frac{3}{4} + dots = 6 + frac{frac{3}{2}}{1-frac{1}{2}} = 6 + 3 = boxed{9}).

$\sum_{n=4}^{\infty} \frac{6}{n^{2}-1}$

Answer

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