As I read in a lecture recently, Richard Hamming described his philosophy of computing with, “the purpose of computing is insight, not numbers.”

On a relatively regular basis, I use numerical methods to verify an analytical solution I’ve reached and occasionally, to reach an analytical result. At one point, I was playing around with the time complexity of heapsort, and I came across a statement about which I was unsure. I verified that it was the case, but it brought about other questions about a more general case.

$$\sum_{h=0}^{\infty}{\frac{h}{n^h}}$$

How does this converge for different $n$’s?

I played around with it for a little bit to see if I saw any pattern that emerged, like with a geometric series or an arithmetic sum, but I came up with nothing. So, I decided to explore the numbers a little bit, writing a short little Ruby script to take the first 100-or-so elements. It turns out that even this approximation provided the insight to get the general expression:

• $$n=2 \Rightarrow 2$$
• $$n=3 \Rightarrow 0.75 = \frac{3}{4}$$
• $$n=4 \Rightarrow 0.4444… = \frac{4}{9}$$

The expression thus seemed to be:

$$\sum_{h=0}^{\infty}{\frac{h}{n^h}} = \frac{n}{(n-1)^2}$$

Of course this needed to be verified, but that task is easy enough. The point is, that sometimes the numerical approximation can give you the exploratory insight needed to solve a problem. Sure, it’s rough around the edges, and isn’t as elegant as I would necessarily like, but it’s very useful.