Ballantine's series is the series for given by

(Borwein and Bailey 2003, pp. 135-136). It has the interesting property that the terms of the second series are decimal shifts of the terms in the first, as observed by Ballantine in 1939.

More things to try:

Weisstein, Eric W. "Ballantine's Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BallantinesSeries.html