What the Limit Comparison Test Is For
Sometimes a series is too complicated to match directly with a geometric series or a -series. The Limit Comparison Test solves this problem by looking at the ratio between your complicated series and a simpler series whose convergence behavior you already know.
The key idea is simple: if two positive-term series have terms that are essentially proportional to each other for large , then they either both converge or both diverge.
For example, consider . The denominator contains both and , but for large , the term dominates. So the series behaves like . The Limit Comparison Test lets us make that intuition precise.
The Formula
Suppose and are positive-term sequences for sufficiently large . If
where , then the two series have the same convergence behavior:
- If converges, then converges.
- If diverges, then diverges.
The number does not need to equal 1. Any finite, positive value works.
How to Choose the Comparison Series
The hardest part is usually not computing the limit. It is choosing a good comparison series.
For rational expressions involving powers of , look at the highest-power terms. Those terms determine the long-run behavior.
- Identify the dominant power of in the numerator and denominator.
- Build a simpler term using those dominant powers.
- Make sure the resulting comparison series is familiar, usually a -series.
- Compute .
- Use the known behavior of the comparison series to conclude convergence or divergence.
This is why the Limit Comparison Test is especially useful for rational functions: lower-order terms become irrelevant as becomes very large.
Worked Examples
Determine whether converges or diverges.
Determine whether converges or diverges.
Determine whether converges or diverges.
Determine whether converges or diverges.
How to Recognize the Right Comparison
For AP Calculus BC problems, the comparison series is often a -series. The most important skill is recognizing what the complicated term behaves like when becomes large.
For instance:
- behaves like .
- behaves like .
- behaves like .
- behaves like .
Limit Comparison vs. Direct Comparison
These two tests have similar goals, but they work differently.
- Direct Comparison Test: You prove an inequality such as and use that inequality to transfer convergence or divergence.
- Limit Comparison Test: You compute the ratio . If the limit is a positive finite number, the two series have the same behavior.
The Limit Comparison Test is often easier when the terms are algebraically complicated because you do not have to construct a precise inequality.
A Fast AP Calculus BC Strategy
- Ignore constants and lower-order terms at first.
- Find the dominant power of .
- Choose a familiar -series as your comparison.
- Calculate .
- Check that the limit is positive and finite.
- Transfer the convergence or divergence conclusion.
Common Mistakes
For rational expressions, the highest power controls the long-run behavior. Do not let a lower-order term such as determine your comparison when a higher power is present.
The standard Limit Comparison Test requires . If the limit is 0 or infinity, the usual theorem does not let you conclude that the two series have the same behavior.
The standard Limit Comparison Test is designed for positive-term series. Make sure and are positive for sufficiently large .
A limit such as is not itself the final answer. You must still state whether the comparison series converges or diverges and then transfer that conclusion to the original series.
Practice Problems
Determine whether converges or diverges using the Limit Comparison Test.
Show answer
Determine whether converges or diverges using the Limit Comparison Test.
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Determine whether converges or diverges using the Limit Comparison Test.
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Suppose and . If converges, what can you conclude about ?
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