Two years later I was taught how to test convergence using Complex Analysis. There was one method for every case. It was pure, simple and beautiful.

I asked (well, demanded to know really) why I wasn't just taught this method from the beginning?!

The answers were:

• clearly you need the practice anyway
• how the tests were developed in response to each new problem is important
• you don't truly appreciate the C::A convergence test until you've used the others extensively
• you could have developed this method yourself two years ago--why'd you wait for us to show you?

The moral of this story is left as an exercise for the reader.