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This looks like a telescopic series to me...Prove that 1/(1*2*3) + 1/(2*3*4) + ... + 1/(n*(n+1)*(n+2)) = 1/4 - 1/(2*(n+1)*(n+2)
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Partial fractions !Hello Erfan,
Are you expected to use partial fraction decomposition or induction, or is the choice of method up to you? Can you show us what you have tried so far?
It is convenient to proceed by induction. Let's suppose that...Prove that 1/(1*2*3) + 1/(2*3*4) + ... + 1/(n*(n+1)*(n+2)) = 1/4 - 1/(2*(n+1)*(n+2)
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Yes it does work! Try it like this: $$\begin{array}{cccccc} S_n = \frac{1/2}1 &- \frac12 &+ \frac{1/2}3 \\ & + \frac{1/2}2 &- \frac13 &+ \frac{1/2}4 \\ && + \frac{1/2}3 &- \frac14 &+ \frac{1/2}5 \\ &&& + \frac{1/2}4 &-\frac15 &+\frac{1/2}6 \\ &&&& + \ldots, \end{array}$$ continuing like that until you get to the row $+\dfrac{1/2}n - \dfrac1{n+1} + \dfrac{1/2}{n+2}.$ Notice that apart from a few terms at the beginning and end of the sum, the terms in each column add up to $0$.Partial fractions !
I tried this 1/(r*(r+1)*(r+2)) = 1/(2r) - 1/(r+1) + 1/(2(r+2))
but it seems that this doesn't work![]()