Details

Snapshot 1: when , the area is filled is

Snapshot 2: when , a square with area is added

Snapshot 3: where and the area filled is very close to 1

The geometric series converges if and only if , and then the sum of the series is . Convergence of can be proven by the integral test, which states that if is continuous, decreasing, and positive, then converges if converges. In this case, , so the integral converges and therefore the geometric series also converges.

Special thanks to the University of Illinois NetMath Program and the mathematics department at William Fremd High School.

References

[1] R. Bayer. "Proof By Picture." (Aug. 13, 2009) www.scribd.com/doc/254948592/Proof-by-Picture.

[2] M. Moody. "Convergence Tests for Infinite Series." (Jun 18, 2013) www.math.hmc.edu/calculus/tutorials/convergence.