Concept preview

What you will be able to do

1. 01State the ratio test: form $L=\lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|$, and conclude absolute convergence if $L<1$, divergence if $L>1$ or $L=\infty$, and no conclusion if $L=1$.
2. 02Compute $L$ for series built from factorials and exponentials by correctly forming and simplifying the ratio $\dfrac{a_{n+1}}{a_n}$.
3. 03Recognise when the ratio test is inconclusive ($L=1$) and know to switch to another test, and interpret $L<1$ as absolute convergence.

Given a series $\sum a_n$, form the limit of the ratio of the size of consecutive terms: $L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|.$ The absolute value is essential — the ratio test measures how fast the term SIZES change, ignoring signs, and it delivers a conclusion about absolute convergence. The number $L$ is the eventual multiplicative shrink (or growth) factor: roughly, far out in the series each term is about $L$ times the previous one.