Comment by srean
I have tingling suspicion that you might have missed the joke.
To date I have not met anyone who thought he summed the terms of the infinite series in geometric series term by term. That would take infinite time. Of course he used the expression for the sum of a geometric series.
The joke is that he missed a clever solution that does not require setting up the series, recognising it's in geometric progression and then using the closed form.
The clever solution just finds the time needed for the trains to collide, then multiply that with the birds speed. No series needed.
Ah. I was going by memory, and I had those two as separate stories. I didn't remember that he said "I did the sum" on the trains problem.