Concept preview

What you will be able to do

1. 01State and apply the DIRECT comparison test: for $0\le a_n\le b_n$, if $\sum b_n$ converges then $\sum a_n$ converges, and if $\sum a_n$ diverges then $\sum b_n$ diverges.
2. 02State and apply the LIMIT comparison test: if $a_n,b_n>0$ and $\lim_{n\to\infty}\dfrac{a_n}{b_n}=L$ with $0<L<\infty$, then $\sum a_n$ and $\sum b_n$ share the same fate.
3. 03Choose an effective comparison series — usually a p-series or a geometric series — by matching the dominant behaviour of the given term for large $n$, and reason correctly about the direction of the inequality.

The DIRECT comparison test rests on a simple picture: a smaller pile of non-negative terms cannot outgrow a larger pile. Suppose $0\le a_n\le b_n$ for all $n$ (beyond some starting index). Then there are two conclusions, and only two. If the larger series $\sum b_n$ CONVERGES, the smaller series $\sum a_n$ is trapped beneath a finite total and must also converge. If the smaller series $\sum a_n$ DIVERGES, the larger series $\sum b_n$ sits above something already infinite and must also diverge. The logic is one-directional, and getting the direction right is everything.