Side Notes

never finished…

Two Series

Cliff Pickover twitted a fun puzzle: Which series is bigger?

The first one is the famous geometric series which sum is equal to 1. The second one seems to be bigger because 1 < n, except for the 0th term, but that 0th term makes a big difference.

First of all, the second series converges. You can prove it by D’Alembert’s theorem

Because the terms are positive, the series converges absolutely, therefore we can rearrange the terms.

Both series are equal.