superconvergence

A sequence $x_0,x_1,\mathrm{\dots }$ superconverges to 0 if, when the $x_i$ are written in base 2, then each number $x_i$ starts with $2^i-1\approx 2^i$ zeroes.For example, the following sequence is superconverging to 0.

In this case it is easy to see that the number of binary 0’s doubles each $x_n$.

A sequence $\{x_i\}$ superconverges to $x$ if $\{x_i-x\}$ superconverges to 0, and a sequence $\{y_i\}$ is said to be superconvergent if there exists a $y$ to which the sequence superconverges.