The Limit Comparison Test is a powerful tool for deciding whether an infinite series converges or diverges. It is especially useful when a series looks complicated but behaves like a simpler known series for large values of n. Instead of comparing terms with inequalities, you compare the long-term ratio of two positive sequences.
This helps turn messy expressions into familiar p-series, geometric series, or harmonic-type series.
The main idea is to choose a comparison series sum b_n whose convergence behavior is already known, then compute lim n to infinity of a_n / b_n. If the limit is a positive finite number L, then sum a_n and sum b_n either both converge or both diverge. The test works because the terms eventually become constant multiples of each other in size.
It does not tell you the sum of the series, only whether the infinite sum converges or diverges.
Key Facts
- Limit Comparison Test: if a_n > 0, b_n > 0, and lim n to infinity of a_n / b_n = L with 0 < L < infinity, then sum a_n and sum b_n have the same convergence behavior.
- A p-series sum 1 / n^p converges when p > 1 and diverges when p <= 1.
- A geometric series sum ar^n converges when |r| < 1 and diverges when |r| >= 1.
- For rational functions of n, compare the highest-power terms in the numerator and denominator.
- Example: sum (3n + 2) / (n^3 + 5) behaves like sum 3n / n^3 = sum 3 / n^2, so it converges.
- If lim n to infinity of a_n / b_n = 0 or infinity, the basic conclusion of the Limit Comparison Test does not automatically apply in both directions.
Vocabulary
- Infinite series
- An infinite series is a sum of infinitely many terms, usually written as sum a_n.
- Convergence
- A series converges if its sequence of partial sums approaches a finite number.
- Divergence
- A series diverges if its partial sums do not approach a finite number.
- Comparison series
- A comparison series is a known series chosen to match the long-term behavior of a more complicated series.
- Limit comparison ratio
- The limit comparison ratio is the limit of a_n / b_n as n approaches infinity.
Common Mistakes to Avoid
- Choosing b_n with unknown behavior, which is wrong because the test only helps if the comparison series is already known to converge or diverge.
- Using the test with negative or sign-changing terms, which is wrong because the standard Limit Comparison Test requires positive terms eventually.
- Concluding from L = 0 that both series behave the same, which is wrong because the usual test only gives the same behavior when 0 < L < infinity.
- Keeping lower-power terms when choosing b_n for rational expressions, which often makes the comparison harder because the largest powers control the behavior as n becomes large.
Practice Questions
- 1 Use the Limit Comparison Test to determine whether sum from n = 1 to infinity of (5n^2 + 1) / (2n^4 + 7) converges or diverges.
- 2 Use the Limit Comparison Test to determine whether sum from n = 2 to infinity of (4n + 3) / (n^2 - 1) converges or diverges.
- 3 Explain why sum from n = 1 to infinity of (n^3 + 2n) / (7n^5 + 1) should be compared to a p-series, and identify the correct value of p.