## 15.6 Application of Power Series: Stirling Formula for n!

The exponential series converges quite rapidly, because of the factorial in
the denominator. It is therefore easy to evaluate this series to any accuracy
on a spreadsheet. You just make a column for j, one for j! one for the j-th
power term, and sum them.

The ratio of the n-th power term to the previous one
is .
This means that the terms increase until the x-th term, and then start downward.
Suppose
x is an integer, say k. Then the largest term is the k-th power one, and it
is .
This term is not the only contribution to exp(k), but it gives a notion of
the order of magnitude of exp(k). The latter is given by multiplied
by the "effective number of terms in the series that contribute to exp(k)". Let
us call the ratio exp(k) divided by,
its largest term, W(k).

We then have

So what is W(k)? If you think about it, you can probably
convince yourself that it increases with k, but not as fast as k itself does.

**Exercise 15.6 I
want you to figure out what it is by computing the ratio on
a spreadsheet for lots of k values.**

Here is a hint: compare your values for
this ratio for k = j and k = 4j. What do you find? This tells you the dependence
of W on k. Then try to find the
exact
limit of W. You can do it!