Question

Sum of factorials to closed formula.
http://www.wolframalpha.com/input/?i=Sum%5Bn%21+%2F+%28n%2B3%29%21%2C+%7Bn%2C0%2Cm%7D%5D

How do I derive that formula, which theorem or definition would help me to translate the sum to the closed form?

Answer 1

i am going to guess partial fractions, since \[\frac{n!}{(n+3)!}=\frac{1}{(n+3)(n+2)(n+1)}\] maybe the sum telescopes when you use partial fractions

Answer 2

If it helps, it looks like:
\[\frac{ 1 }{ 1*2*3 } + \frac{ 1 }{ 2*3*4 } + ... + \frac{ 1 }{ m*(m-1)*(m-2) }\]

Answer 3

Thanks satellite73, I'll look it up

Answer 4

yeah it is partial fractions and ugly algebra