Root and Ratio Tests

Root & Ratio Tests — This video covers the two workhorse convergence tests for infinite series. Given a series Σaₙ, we examine the limiting behavior of ⁿ√|aₙ| (root test) and |aₙ₊₁/aₙ| (ratio test): if the limit is less than 1 the series converges (absolutely), if greater than 1 it diverges, and if it equals 1 the test is inconclusive. We'll prove both tests by comparison with geometric series, see why the root test is strictly stronger than the ratio test, and work through examples — including the inconclusive boundary cases where L = 1.