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!