5
$\begingroup$

I was playing around with factorials on Wolfram|Alpha, when I got this amazing result :

$$\sum \limits_{n=0}^{\infty} \dfrac{n!}{(n+1)!+(n+2)!} = \dfrac{3}{4}.$$

Evaluating the first few partial sums makes it obvious that the sum converges to $\approx 0.7$. But I am not able to prove this result algebraically. I tried manipulating the terms and introducing Gamma Function, but without success.

Can anyone help me with this infinite sum ? Is there some well-known method of evaluating infinite sums similar to this ?

Any help will be gratefully acknowledged.

Thanks in advance ! :-)

EDIT : I realized that $(n!)$ can be cancelled out from the fraction and the limit of the remaining fraction as $n \to \infty$ can be calculated very easily to be equal to $0.75$. Very silly of me to ask such a question !!! Anyways you can check out the comment by @Did if this "Edit" section does not help.

  • 3
    I definitely haven't tried, but I wonder if you can factor $(n+1)!$ out of the bottom and write it as a pair of fractions so that the sum telescopes. – 2017-02-17
  • 15
    Perhaps not quite so amazing... $$\frac{n!}{(n+1)!+(n+2)!}=\frac1{(n+1)(n+3)}=\frac12\left(\frac1{n+1}-\frac1{n+3}\right)$$ hence the $n$th sum *telescopes* into $$\sum_{k=0}^n\frac{k!}{(k+1)!+(k+2)!}=\frac12\left(\frac11+\frac12-\frac1{n+2}-\frac1{n+3}\right)$$ which converges to $$\frac34$$ – 2017-02-17
  • 1
    @Did Thanks for the useful comment ... but it is still amazing for me :) – 2017-02-17
  • 1
    Make that an answer. – 2017-02-17
  • 0
    Some would upvote a comment and wont for an answer. – 2017-02-18
  • 0
    @ZaidAlyafeai Well, some were here long before this answer was posted, and others don't feel like the reproduction of work done in comments by someone else is a good thing, since you kinda are taking their efforts. But I see no problem, since I made the answer community wiki, since it rightly came from the community and not I, which means it will not impact my rep or anything like that. $\ddot\smile$ – 2017-02-18

1 Answers 1

9

Thanks to pjs36 and Did,

Notice that:

$$\begin{align}a_n&=\frac{n!}{(n+1)!+(n+2)!}\\&=\frac1{(n+1)+(n+1)(n+2)}\\&=\frac1{(n+1)(n+3)}\\&=\frac12\left(\frac1{n+1}-\frac1{n+3}\right)\end{align}$$

Thus, we get a telescoping series, leaving us with:

$$\sum_{n=0}^\infty\frac{n!}{(n+1)!+(n+2)!}=\frac12\left(\frac1{0+1}+\frac1{1+1}\right)=\frac34$$