Series Convergence Tests

The nth-term, p-series, and ratio tests.

The idea

A convergence test is a checkable condition on the terms of a series that carries a guaranteed verdict. The series $\sum_{n=1}^{\infty} a_{n}$ means the limit of its partial sums $a_{1} + a_{2} + \cdots + a_{N}$ as $N$ grows; whether that limit exists is rarely visible from the terms directly, so we apply a test. The first test comes first because it costs the least.

Theorem (The nth-term test).

If $a_{n}$ does not tend to $0$, then $\sum a_{n}$ diverges.

The test can only reject. Terms that do shrink to $0$ prove nothing, and the harmonic series $\sum \tfrac{1}{n}$ is the standard warning: its terms tend to $0$ while its partial sums grow past every bound.

Theorem (The p-series test).

The series $\sum_{n=1}^{\infty} \dfrac{1}{n^{p}}$ converges when $p > 1$ and diverges when $p \le 1$.

The harmonic series is the borderline case $p = 1$. Plotted side by side, the partial sums of $\sum \tfrac{1}{n^{2}}$ flatten toward their limit while the harmonic sums climb past every bound.

The last test suits terms built from powers and factorials, where each term is roughly a fixed multiple of the one before.

Theorem (The ratio test).

Let $L = \lim_{n \to \infty} \left| \dfrac{a_{n+1}}{a_{n}} \right|$. If $L < 1$ the series $\sum a_{n}$ converges; if $L > 1$ it diverges; if $L = 1$ the test gives no verdict.

When the ratio test is silent, another test must decide.

Ways to work on it

Not sure where to start? Take the ten-question placement test.