> this simple one still converges at about 0.6 decimal digits per term.<p>Quick proof of this: as the number of terms <i>n</i> in the sum goes to infinity, the ratio of each term to the previous one is approximately 1/4 - the first factor contributes <i>m/(m+1)</i>, the second <i>q/(q+2)</i> for some <i>m</i> and <i>q</i> that go to infinity along with <i>n</i>, the third contributes 1/4.<p>If we counted base 4, then the value of each digit would be on average 1/4 of the previous one, certainly for a normal number like pi. But we count base 10, so we get <i>log_10 4</i> decimal digits every time we get one base-four digit. Which is very close to 0.6.