Choosing a Convergence Test
A small toolkit of tests turns the vague question of whether a sum settles down into a routine you can run in a few lines.
Divergence test first
Always start here: if the terms a_n do not approach 0, the series must diverge. For example, the terms of n divided by (n+1) approach 1, so that series diverges immediately. The catch is that the test is one-way, since the harmonic series has terms approaching 0 and still diverges.
\lim_{n \to \infty} a_n \neq 0 \implies \sum a_n \text{ diverges}Ratio and comparison
The ratio test computes the limit of the absolute ratio of consecutive terms, call it L, and converges when L is below 1. For the series with terms n^2 divided by 2^n, the ratio limit is 1/2, so it converges. Comparison tests let you borrow a verdict from a simpler series with terms of similar size.
L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|p-series benchmark
A p-series adds up 1 over n to the power p, and it converges exactly when p is greater than 1. The harmonic series is the p-series with p equal to 1, and the integral test shows its totals grow like the natural log, so it diverges. Memorizing the p-series cutoff gives you a reliable yardstick for comparison.
\sum_{n=1}^{\infty} \frac{1}{n^p} \text{ converges iff } p > 1Flashcard
Term test forces divergence of the series
Check your understanding
What does the ratio test tell you about the series whose nth term is n^2 divided by 2^n?