Convergence Tests

Level 1 - Math I (Physics) topic page in Series.

Principle

Convergence tests decide whether a non-negative infinite series has a finite sum even when that sum is not known. They compare the tail of the series with behavior that is already understood.

This is often enough for physics models: knowing that an approximation stabilizes can be more important than knowing the exact limiting value.

Notation

\(a_n\) is the non-negative term of the series being tested.
\(b_n\) is the term of a comparison series.
\(\rho\) is the limiting value used in the ratio or root test.
A comparison series is a known series used to bound or compare \(\sum a_n\).
The ratio test studies \(a_{n+1}/a_n\).
The root test studies \(a_n^{1/n}\).
The integral test compares \(\sum a_n\) with an improper integral of a positive decreasing function.

Method

Use a decision sequence rather than trying tests at random.

Check the term limit. If \(a_n\) does not tend to \(0\), the series diverges.
Compare with a known geometric series or \(p\)-series when the terms have similar size.
Try the ratio test for factorials, exponentials, and powers multiplied together.
Try the root test when the whole term is raised to a power depending on \(n\).
Try the integral test for terms coming from functions like \(1/n^q\), after checking positivity and decreasing behavior.

Ratio test

\[\rho=\lim_{n\to\infty}\frac{a_{n+1}}{a_n}\]

Root test

\[\rho=\lim_{n\to\infty}a_n^{1/n}\]

p-series result

\[\sum_{n=1}^{\infty}\frac{1}{n^q}\text{ converges exactly when }q\gt1\]

Rules

Comparison test: if \(0\le a_n\le b_n\) eventually and \(\sum b_n\) converges, then \(\sum a_n\) converges.
Comparison test for divergence: if \(a_n\ge b_n\ge0\) eventually and \(\sum b_n\) diverges, then \(\sum a_n\) diverges.
Limit comparison: if \(\lim_{n\to\infty}a_n/b_n=L\) with \(0\lt L\lt\infty\), then \(\sum a_n\) and \(\sum b_n\) have the same convergence behavior.
Ratio test: if \(\rho\lt1\), the series converges; if \(\rho\gt1\), it diverges; if \(\rho=1\), the test is inconclusive.
Root test: if \(\rho\lt1\), the series converges; if \(\rho\gt1\), it diverges; if \(\rho=1\), the test is inconclusive.
Integral test: if \(a_n=f(n)\) where \(f\) is positive and decreasing, then \(\sum a_n\) converges exactly when the corresponding improper integral has a finite limit.

Examples

For \(\sum_{n=1}^{\infty}n^2/2^n\), use the ratio test because powers of \(n\) and exponentials appear together.

Ratio example

\[\frac{a_{n+1}}{a_n}=\frac{(n+1)^2}{2^{n+1}}\frac{2^n}{n^2}=\frac12\left(1+\frac1n\right)^2\to\frac12\]

The limit is \(\rho=1/2\lt1\), so the series converges.

For a \(p\)-series \(\sum_{n=1}^{\infty}1/n^q\), the integral test uses \(f(x)=1/x^q\). The result is convergence exactly for \(q\gt1\) and divergence for \(q\le1\).

Checks

Verify that terms are non-negative before using positive-series comparison, ratio, root, or integral tests in this form.
Interpret \(\rho=1\) as inconclusive, not as convergence.
For the integral test, check that \(f\) is positive and decreasing on the tail of the series.
Match the test to the term structure: comparisons for similar size, ratio for factorials and exponentials, root for powers of \(n\).

Positive Series

Negative Terms