Using the Series Comparison Tests for Convergence

(Video) Limit Comparison Test and Direct Comparison Test

by PatrickJMT

Here's a video by patrickJMT showing you how to use the limit comparison test to see if a series converges.

Overview of Applying the Series Comparison Tests

The motivating idea behind series comparison tests is to compare a complicated series to a simpler one whose convergence/divergence is easier to determine. Some common candidates for the "simpler series" are infinite geometric series (the idea behind the root test and ratio test) and $p$-series. It takes some practice to identify a good candidate to choose, in order to get a conclusive test result without too much effort. (Sometimes if the choice is too simple, we get an inconclusive comparison, because we have lost too much information in the simplification. For example, comparing $\frac11-\frac12+\frac13-\frac14\pm\dots$ to $\frac11+\frac12+\frac13+\frac14+\dots$ wouldn't work: the former converges by the alternating series test, while the latter diverges by the $p$-series test with $p=1$.)