Series Convergence Tests & Power Series

Geometric, p-series, ratio test, and the radius of convergence.

The idea

An infinite series $\sum_{n=1}^{\infty} a_{n}$ adds infinitely many numbers, and its sum is defined through partial sums: let $s_{N} = a_{1} + a_{2} + \cdots + a_{N}$. If the sequence $s_{1}, s_{2}, s_{3}, \ldots$ approaches a finite limit, the series converges and that limit is its sum; otherwise it diverges.

The limit can rarely be computed directly, so the working question is whether a series converges at all. We answer it by comparing the series with a family whose behavior is already settled, and two families settle most cases. A geometric series $\sum r^{n}$, whose terms shrink by a constant factor, converges if and only if $|r| < 1$. A $p$-series $\sum 1/n^{p}$, whose terms shrink only polynomially, converges if and only if $p > 1$. The two verdicts are visible in the partial sums themselves: for $\sum 1/n^{2}$ they settle onto a limit, while for the borderline harmonic series $\sum 1/n$ (the case $p = 1$) they climb past every bound, however slowly.

The ratio test carries out the geometric comparison automatically. Put $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_{n}} \right|.$ When $L < 1$ the terms eventually shrink like a geometric series of ratio $L$, so the series converges absolutely; when $L > 1$ the terms eventually grow, so it diverges. When $L = 1$ the test decides nothing, and another test must settle the question.

Ways to work on it

• Walkthrough. Convergence tests: geometric, p-series, and the ratio test.
• Practice. Classify a series as convergent or divergent.
• Hardest. Find a power series' radius of convergence by the ratio test.

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