Separating a Factorial Expression into a Telescoping Sequence

In my Calculus 2 course, we were asked to make the expression:
a_n = \frac{1}{(n + k)!}

into a telescoping sequence. In the past we have done things like:
\frac{1}{2^n} = \frac{1}{2^{n-1}} – \frac{1}{2^n}

What I am trying to figure out is how to get the uppermost expression to look like the one just above. It is an infinite sum of an with n being the variable and k being some fixed natural number. Any input would be greatly appreciated.



2 Answers


Something like this?

1n!=n+1(n+1)!=n(n+1)!+1(n+1)!1(n+1)!=1n!−n(n+1)!1(n+1)!=1n!−(n+1)(n−1)!\frac {1}{n!} = \frac {n+1}{(n+1)!} = \frac {n}{(n+1)!} + \frac {1}{(n+1)!}\\
\frac {1}{(n+1)!} = \frac {1}{n!} – \frac {n}{(n+1)!}\\
\frac {1}{(n+1)!} = \frac {1}{n!} – \frac {(n+1)}{(n-1)!}\\



How would k be implemented?
– E. Wright
Oct 20 at 20:48

\frac1{(n+k)!} &= \sum_{j=n+k}^\infty\left( \frac1{j!}-\frac1{(j+1)!}\right)