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I need some help with the following question from my homework, I do not exactly understand what to do. Question at Hand

Text is: Evaluate the partial sums of the infinite series $\displaystyle \sum_{n=1}^\infty \frac1{n(n+2)}$, and then evaluate the infinite series.

The trouble I am having is understanding exactly what is asked of me to do.

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Well, you try to find an expression for the partial sums first. Do you know partial fraction decomposition? – J. M. Oct 19 '11 at 1:30
The question is asking you to find a formula for $S(M) = \sum_{n \leq M} \frac{1}{n(n+2)}$. – JavaMan Oct 19 '11 at 1:31
up vote 4 down vote accepted

The m-th partial sum of $ \displaystyle \sum_{n=1}^{\infty} \frac{1}{n(n+2) } $ is the sum truncated to the m-th term. In other words, it first wants you to find the finite sum $$ s_m = \sum_{n=1}^m \frac{1}{n(n+2)} $$ for all $m\in \mathbb{N}$. Then it wants you to find the original infinite sum by recalling the definition that $$ \sum_{n=1}^{\infty} \frac{1}{n(n+2) } = \lim_{m\to \infty} s_m .$$

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@Steve: And to find $s_m$, you’ll want to use partial fractions. After you’ve found the partial fraction decomposition, you may also find it very helpful to write out the first few partial sums in full, paying close attention to the algebraic signs of the terms. – Brian M. Scott Oct 19 '11 at 1:53

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