I recently came across a complicated sum while working on my homework. Usually I have no issues evaluating sums, but this one has stumped me. WolframAlpha managed to find a closed form solution, but I can't seem to work out how I should go about deriving it. Here's the sum and closed form from WolframAlpha:


I have tried writing out the sum term by term to look for patterns. I noticed the factorials in the denominator seem to pair up sometimes, but I haven't been able to leverage that to any use. The main way I know to handle factorials in sums is to find a power series representation that matches the sum, but this sum doesn't seem to match anything I can find.

I appreciate any hints or solutions. Ideally, a solution without any prior knowledge of the result.

  • 1
    $\begingroup$ Hint : If you multiply the sum with $(y-1)!$ , you can use the binomial theorem because the binomial coeffcient $\binom {y-1}{x-1}$ appears. $\endgroup$ – Peter Apr 28 '18 at 7:48
  • $\begingroup$ @Peter thanks! I was just forgetting that the binomial theorem exists. $\endgroup$ – superckl Apr 28 '18 at 7:58

These should probably be enough.

Hint 1: Use the binomial theorem.

Hint 2: $\dfrac{1}{(x - 1)!(y - x)!} = \dfrac{1}{(y - 1)!}\displaystyle{y - 1 \choose x - 1}$

Hint 3: $\dfrac{11}{6} = 1 + \dfrac{5}{6}$

  • $\begingroup$ Thanks, I was just missing the binomial theorem. $\endgroup$ – superckl Apr 28 '18 at 7:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for?Browse other questions tagged or ask your own question.