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I have a question regarding the sum of this series: $$\frac{3}{1!+2!+3!} + \frac{4}{2!+3!+4!} + \cdots + \frac{2008}{2006!+2007!+2008!}$$ My approach: I found that this sum is equal to: $$\sum_{n=3}^{2008}\frac{n}{(n-2)!+(n-1)!+(n)!}$$

I reduced it to : $$\sum_{n=3}^{2008}\frac{1}{n(n-2)!}$$
Please suggest how to proceed further.

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What we want is $$\sum_{n=1}^{N} \dfrac{n+2}{n! + (n+1)! +(n+2)!}$$ \begin{align} \dfrac{n+2}{n! + (n+1)! +(n+2)!} & = \dfrac{n+2}{n! \left( 1 + (n+1) + (n+1)(n+2) \right)}\\ & = \dfrac{n+2}{n! \left( n^2 + 4n + 4 \right)}\\ & = \dfrac1{n! \left( n+2 \right)}\\ & = \dfrac{n+1}{(n+2)!}\\ & = \dfrac{n+2}{(n+2)!} - \dfrac1{(n+2)!}\\ & = \dfrac1{(n+1)!} - \dfrac1{(n+2)!} \end{align}

Can you finish it off from here?

Move your mouse over the gray area below for the complete answer.

\begin{align}\sum_{n=1}^{N} \dfrac{n+2}{n! + (n+1)! +(n+2)!} & = \sum_{n=1}^{N} \left( \dfrac1{(n+1)!} - \dfrac1{(n+2)!}\right)\\ & = \left( \dfrac1{2!} - \dfrac1{3!} + \dfrac1{3!} - \dfrac1{4!} + \dfrac1{4!} - \dfrac1{5!} + \cdots + \dfrac1{(N+1)!} - \dfrac1{(N+2)!}\right)\\ & = \dfrac1{2!} - \dfrac1{(N+2)!}\end{align} Set $N=2006$ to get the answer to your question.

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  • $\begingroup$ What should $n$ replaced by ? $\endgroup$ – spa Sep 26 '12 at 17:16
  • $\begingroup$ @spa $N = 2006$ $\endgroup$ – user17762 Sep 26 '12 at 17:16
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Hint: Apply telescoping series to $$ \begin{align} \sum_{k=1}^{2006}\frac{k+2}{k!+(k+1)!+(k+2)!} &=\sum_{k=1}^{2006}\frac{k+2}{k!\,(k+2)^2}\\ &=\sum_{k=1}^{2006}\frac{k+1}{(k+2)!}\\ &=\sum_{k=1}^{2006}\left(\frac1{(k+1)!}-\frac1{(k+2)!}\right) \end{align} $$

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  • $\begingroup$ elegant and concise $\endgroup$ – G Cab Feb 12 '18 at 14:39
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$$\dfrac1{n(n-2)!}=\dfrac{n-1}{n!}$$ which is Telescoping

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  • $\begingroup$ Thanks, how did you telescope it? $\endgroup$ – drake01 Feb 12 '18 at 13:26
  • $\begingroup$ Drake write out a few terms, they start cancelling. $\endgroup$ – King Tut Feb 12 '18 at 13:29
  • $\begingroup$ Yes, they start cancelling out when I write out a few terms of RHS, but how do you convert LHS to RHS( using partial fractions?) $\endgroup$ – drake01 Feb 12 '18 at 13:32
  • $\begingroup$ @drake01 I'd say it comes by practice and fiddling out with the few terms given. You know this is not a standard sequence. So, the next obvious way is to try see if the sequence can telescope. You try out a few techniques, multiply $(n-1)$ in both the numerator and denominator, and it works! As noted on MSE before, call it the maths of "wishful thinking" ^_^ $\endgroup$ – Gaurang Tandon Feb 12 '18 at 13:35
  • $\begingroup$ You are right, @Gaurang, $\endgroup$ – drake01 Feb 12 '18 at 13:38
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$\sum_{n=3}^{2008}\frac{1}{n(n-2)!}=\sum_{n=3}^{2008}\frac{1}{n(n-2)!}.\frac{(n-1)}{(n-1)}=\sum_{n=3}^{2008}\frac{n-1}{n!}=\sum_{n=3}^{2008}(\frac{n}{n!}-\frac{1}{n!})=\sum_{n=3}^{2008}(\frac{1}{(n-1)!}-\frac{1}{n!})=\frac{1}{2!}-\frac{1}{3!}+\frac{1}{3!}-\frac{1}{4!}+...-\frac{1}{2007!}+\frac{1}{2007!}-\frac{1}{2008!}=\frac{1}{2!}-\frac{1}{2008!}$

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  • $\begingroup$ The matter of fun is , I had obtained $\sum\frac{n-1}{n!}$ earlier then converted it into,$\sum\frac{1}{n(n-2)!}$ and then posted in MSE. Because the concept of Telescoping series didn't struck me at the moment. btw thanks. $\endgroup$ – drake01 Feb 12 '18 at 13:52
  • $\begingroup$ @drake01 We all gradually gain the experience. The sweet part is these things, yes so funny. LOL :D $\endgroup$ – Mehrdad Zandigohar Feb 12 '18 at 13:57

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