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I was thinking about this problem:

Given a series $\sum_{n=4}^{\infty}\frac{1}{n^{2}-1}$ ,how can i show that its sum is a/an rational/irrational number,given that the series converges?

Could someone point me in the right direction? Thanks everyone in advance.

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    Partial fractions.2012-12-02

2 Answers 2

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$1/(n-1)-1/(n+1) = 2/(n^2-1) $, so $\sum_{n=4}^{\infty} \frac{1}{n^2-1} = (1/2)\sum_{n=4}^{\infty} \left(\frac{1}{n-1} - \frac{1}{n+1}\right) = \frac{1}{3} + \frac{1}{4} $ since all the later terms are cancelled out.

Whoops - as pointed out by Limitless, this should be $\frac{1}{2}\left(\frac{1}{3} + \frac{1}{4}\right)$.

Note that this also allows you to get an explicit expression for $\sum_{n=a}^b \frac{1}{n^2-1}$ for any integers $a$ and $b$.

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    Thanks for correcting my oversight. I will edit my answer with acknowledgement.2012-12-02
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Find the closed form and take the limit in the case of infinite sums. That is, you find the closed form of the sum $\sum_{4 \le k \le m}\frac{1}{k^2-1}$ and evaluate $\lim_{m \to \infty}(\text{closed form of the sum})$.

In this case, you apply partial fraction decomposition to $\frac{1}{k^2-1}$ and arrive at $\frac{1}{k^2-1}=\frac{-\frac{1}{2}}{k+1}+\frac{\frac{1}{2}}{k-1}=\frac{1}{2}\left(\frac{1}{k-1}-\frac{1}{k+1}\right).$

Taking the partial sum $\sum_{4 \le k \le m}\frac{1}{k^2-1}=\frac{1}{2}\sum_{4 \le k \le m}\frac{1}{k-1}-\frac{1}{k+1}.$

The sum telescopes, so we see that $\sum_{4 \le k \le m}\frac{1}{k+1}-\frac{1}{k-1}=\color{red}{\frac{1}{3}}-\color{green}{\frac{1}{5}}+\color{red}{\frac{1}{4}}-\frac{1}{6}+\color{green}{\frac{1}{5}}-\frac{1}{7}+\dots+\color{green}{\frac{1}{m-1}}-\frac{1}{m-3}+\color{red}{\frac{1}{m}}-\frac{1}{m-2}+\color{red}{\frac{1}{m+1}}-\color{green}{\frac{1}{m-1}}=\frac{1}{3}+\frac{1}{4}+\frac{1}{m}+\frac{1}{m+1}.$

Taking the limit, we have $\lim_{m \to \infty}\frac{1}{3}+\frac{1}{4}+\frac{1}{m}+\frac{1}{m+1}=\frac{1}{3}+\frac{1}{4}+\lim_{m \to \infty}\frac{1}{m}+\frac{1}{m+1}=\frac{1}{3}+\frac{1}{4}=\frac{7}{12}.$

Recalling the factor of $\frac{1}{2}$, we arrive at $\sum_{4 \le k \le \infty}\frac{1}{k^2-1}=\frac{1}{2}\cdot\frac{7}{12}=\frac{7}{24}$.

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    @AntonioVargas, LOL. What luck I have! I will keep that in mind. :)2012-12-03