Comparison Tests for Convergence

Why Comparison Tests Work

When you are given an infinite series, you often do not need to find its exact sum to determine whether it converges. Instead, you can compare it with a series whose behavior you already know.

The basic idea is simple: if the terms of a positive series are smaller than the terms of a known convergent series, the original series must also converge. Conversely, if the terms are larger than the terms of a known divergent series, the original series must also diverge.

The key is choosing a good comparison series. The most useful benchmark series are usually geometric series and p-series.

The Direct Comparison Test

Suppose and are nonnegative for sufficiently large . If:

then the comparison tells you exactly what you need to know:

• If converges, then also converges.
• If diverges, then also diverges.

Notice that the two directions are different. A smaller series being smaller than a convergent series guarantees convergence. A larger series being larger than a divergent series guarantees divergence.

The Comparison Test in Plain English

Think of a convergent series as a "finite bucket." If your positive terms are always smaller than the terms going into that finite bucket, your series cannot overflow.

Think of a divergent series as an "overflowing bucket." If your positive terms are at least as large as terms that already overflow, your series must also overflow.

How to Choose a Comparison Series

The hardest part of the Direct Comparison Test is usually not the test itself. It is finding a useful series to compare against.

Look for the dominant behavior of the terms as becomes large. Ignore constants and lower-order terms when identifying what controls the behavior.

Power-like expressions

If the denominator behaves approximately like , a p-series is usually a natural comparison:

Geometric behavior

If the terms contain a constant raised to the power, such as , look for a geometric series:

Worked Examples: Direct Comparison

The Limit Comparison Test

Sometimes you know that two series behave similarly, but it is difficult to prove a simple inequality between their individual terms. That's where the Limit Comparison Test is especially useful.

For positive-term series, calculate the limit of the ratio of the terms:

If the limit is a finite positive number, the two series have the same convergence behavior:

So if is known to converge, also converges. If diverges, also diverges.

Why the Limit Comparison Test Is Powerful

The limit comparison test essentially says that multiplying the terms of a series by a positive constant does not change whether the series converges or diverges.

If the ratio approaches a number such as , , or , the two series have the same long-term behavior.

That means you can often strip away complicated details and compare only the highest-power terms.

Worked Examples: Limit Comparison

Direct Comparison vs. Limit Comparison

Common Mistakes

A Fast AP Calculus BC Strategy

1. Look at the terms. Ask what the expression behaves like for very large .
2. Find a familiar benchmark. Usually this is a geometric series or a p-series.
3. Try Direct Comparison if a clean inequality is obvious.
4. Try Limit Comparison if the dominant terms clearly match but the inequality is messy.
5. State the known result. Explain whether your comparison series converges or diverges.
6. State the conclusion. Explicitly name the test and whether the original series converges or diverges.

Practice Problems