A Converging Geometric Series
This Demonstration shows that .[more]
The square is half empty with area when . The square is empty at . In general, the square is empty at step and full, which shows that converges to 1.[less]
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.
 R. Bayer. "Proof By Picture." (Aug. 13, 2009) www.scribd.com/doc/254948592/Proof-by-Picture.
 M. Moody. "Convergence Tests for Infinite Series." (Jun 18, 2013) www.math.hmc.edu/calculus/tutorials/convergence.