By the definition above,
and by substitution we get
If we keep following this procedure, we'll start to see a pattern:
and so on, down to
Now,
One of these series is geometric and has a well-known closed form:
The other (see note below) is a bit less obvious, but can be derived:
so we have
# # #
A brief note on how to compute the sum,
The first term of the sum is 0:
Add and substract 1 to the first factor in the summand and expand the sum into two sums:
The latter sum we're familiar with. For the other sum, we can shift the index to make it start at , and again the first term in the sum would be 0:
So we have
Continuing in this way, we would end up getting
or as a double sum,
which is easy to reduce: