Comparison Tests

Summary

Comparison tests decide convergence of series with nonnegative terms by relating them to a known series. The direct comparison test uses inequalities; the limit comparison test uses asymptotic ratios.

Prerequisites

P Series, Geometric Series, Harmonic Series

Theorems

Direct comparison

Assume $0\le a_n\le b_n$ for all $n$ large.

• If $\sum b_n$ converges, then $\sum a_n$ converges.
• If $\sum a_n$ diverges, then $\sum b_n$ diverges.

Limit comparison

If $a_n>0$ , $b_n>0$ , and

then $\sum a_n$ and $\sum b_n$ both converge or both diverge.

Worked Example

Correct comparison for $1/(n(n+1))$

For $n\ge 1$ , $0<{\displaystyle \frac{1}{n(n+1)}}<{\displaystyle \frac{1}{n^2}}$ . Since $\sum 1/n^2$ converges ( $p=2$ ), so does $\sum 1/(n(n+1))$ .

Alternatively, partial fractions show the series telescopes to $1$ .

False claim to avoid

The inequality ${\displaystyle \frac{1}{n(n+1)}}<{\displaystyle \frac{1}{2^n}}$ is not true for all large $n$ . Polynomial decay is slower than exponential decay, so $1/(n(n+1))\gg 1/2^n$ as $n\to \mathrm{\infty }$ . Do not compare this series to a geometric series via a false bound.

Limit comparison

For $a_n=1/(n^2+n)$ and $b_n=1/n^2$ , $a_n/b_n\to 1$ , so $\sum a_n$ converges.

Common Mistakes

• Comparing with the wrong direction of the inequality.
• Using a false geometric bound for rational terms.
• Applying comparison tests to series with negative terms without absolute values.

Connections

• P Series, Integral Test, Absolute Convergence

References

Comparison tests are in OpenStax Calculus Volume 2.^[1]