Series Convergence Tests — Ratio, Root, Comparison & More

Deciding whether an infinite series converges or diverges is rarely about memorizing every test. It is about matching the shape of the terms to the one test that settles the question fastest. This page organizes the standard tests around that idea.

The decision shortcut

When you face a series, run through this order:

1. Check whether $a_n \to 0$. If it does not, the series diverges.
2. Look for a known pattern first, especially geometric series or $p$-series.
3. Use comparison for positive terms that resemble a familiar benchmark.
4. Use ratio or root when factorials, exponentials, or powers dominate.
5. Use the alternating series test only when signs alternate and term sizes decrease to $0$.

For a series

$$\sum_{n=1}^{\infty} a_n,$$

convergence means the partial sums approach a finite limit; divergence means they do not. Note that a convergence test usually does not compute the sum. It only classifies. That distinction matters because the goal is almost always classification, not evaluation.

Why each test fits its shape

The term test comes first because it is cheap

If

$$\lim_{n \to \infty} a_n \ne 0,$$

then

$$\sum_{n=1}^{\infty} a_n$$

diverges. This $n$th-term test is one-way only: $a_n \to 0$ does not guarantee convergence. The harmonic series

$$\sum_{n=1}^{\infty} \frac{1}{n}$$

diverges even though $\frac{1}{n} \to 0$.

Geometric and $p$-series are the reference shapes

A geometric series

$$\sum_{n=0}^{\infty} ar^n$$

converges when $|r| < 1$ and diverges when $|r| \ge 1$. A $p$-series

$$\sum_{n=1}^{\infty} \frac{1}{n^p}$$

converges when $p > 1$ and diverges when $p \le 1$. Recognizing these first is what makes comparison work, because they are the benchmarks you compare against.

Comparison and limit comparison match rational shapes

Use the comparison test for positive terms, and the logic is intuitive: if your terms are no larger than the terms of a known convergent series, your series also converges; if your terms are at least as large as a known divergent series, yours also diverges. Because it rests on inequalities, it is cleanest when you can bound terms directly. When clean inequalities are awkward but two positive-term series share the same dominant behavior, use limit comparison instead. For

$$a_n > 0, \qquad b_n > 0,$$

if

$$\lim_{n \to \infty} \frac{a_n}{b_n} = c$$

for finite $c > 0$, then $\sum a_n$ and $\sum b_n$ both converge or both diverge. This is usually the cleanest choice for rational expressions in $n$.

Ratio and root match factorials, exponentials, and $n$th powers

Use the ratio test when factorials or exponential factors appear, because those grow or shrink by a roughly constant factor each step, which is exactly what a ratio measures:

$$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|.$$

If $L < 1$, the series converges absolutely; if $L > 1$ or $L = \infty$, it diverges; if $L = 1$, the test is inconclusive, so that case settles nothing on its own. Use the root test when an $n$th power is built in, like terms of the form $(\cdots)^n$, since taking the $n$th root cancels that power cleanly:

$$L = \lim_{n \to \infty} \sqrt[n]{|a_n|},$$

with the same three conclusions as the ratio test.

Alternating and integral tests have strict conditions

For alternating signs, in a form like

$$\sum (-1)^n b_n \quad \text{or} \quad \sum (-1)^{n+1} b_n,$$

with $b_n \ge 0$ eventually decreasing and $b_n \to 0$, the series converges, though not necessarily absolutely. That gap between convergence and absolute convergence is exactly the difference between conditional and absolute convergence, so the alternating series test alone never certifies absolute convergence.

The integral test applies when the series comes from a positive, continuous, decreasing $f(x)$ with $f(n) = a_n$ for large $n$; then

$$\sum_{n=1}^{\infty} a_n \quad \text{and} \quad \int_1^{\infty} f(x)\,dx$$

both converge or both diverge. It is especially useful for logarithmic and power-based terms, but only when positivity, continuity, and the decreasing condition all hold on the interval you use.

Worked example: ratio test on $\sum \frac{n}{2^n}$

Consider

$$\sum_{n=1}^{\infty} \frac{n}{2^n}.$$

The exponential factor $2^n$ points to the ratio test. With $a_n = \frac{n}{2^n}$,

$$\left| \frac{a_{n+1}}{a_n} \right| = \frac{(n+1)/2^{n+1}}{n/2^n} = \frac{n+1}{2n},$$

and

$$\lim_{n \to \infty} \frac{n+1}{2n} = \frac{1}{2}.$$

Because $\frac{1}{2} < 1$, the series converges absolutely. The real lesson is the choice of test: the $2^n$ makes the ratio simplify cleanly, so the ratio test wins with little algebra.

Practice picking the test

Try

$$\sum_{n=1}^{\infty} \frac{3^n}{n!}.$$

Before any algebra, decide which test fits the shape and say why. Then solve it, and ask whether the same test would still be your first pick for

$$\sum_{n=1}^{\infty} \frac{n^2}{3^n}.$$

Comparing the two cements the pattern.

Calculation traps to avoid

Using a test that does not match. Rational functions of $n$ favor comparison or limit comparison; factorials and exponentials favor ratio.

Forgetting the conditions. Comparison tests need positive terms. The alternating series test needs eventually decreasing positive magnitudes with limit $0$. The integral test needs positivity, continuity, and decreasing behavior.

Treating $L = 1$ as a verdict. For ratio and root tests, $L = 1$ means the test failed to decide. Use a different approach.

Assuming $a_n \to 0$ is enough. It is necessary, not sufficient. The harmonic series is the standard counterexample.

Where these tests matter

Convergence tests run throughout calculus and analysis: classifying infinite sums, justifying power-series manipulations, and checking whether an approximation is mathematically safe. The underlying skill is pattern recognition, matching a series to the test that exposes its structure fastest.