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.