The Ratio Test is a powerful method for deciding whether an infinite series converges or diverges. It compares the size of consecutive terms by taking the limit of |a_{n+1}/a_n| as n grows without bound. This test matters because many important series contain factorials, powers, or products that are difficult to analyze by simpler tests.
A single limit often gives a clear answer about the long-term behavior of the series.
The key idea is to measure whether the terms shrink fast enough from one term to the next. If the limiting ratio L is less than 1, the terms eventually decrease like a convergent geometric series, so the series converges absolutely. If L is greater than 1 or infinite, the terms do not shrink fast enough, so the series diverges.
If L = 1, the Ratio Test gives no conclusion, and another test is needed.
Key Facts
- Ratio Test limit: L = lim as n -> infinity |a_{n+1}/a_n|.
- If L < 1, then the series sum a_n converges absolutely.
- If L > 1 or L = infinity, then the series sum a_n diverges.
- If L = 1, the Ratio Test is inconclusive.
- The Ratio Test is especially useful for terms with factorials, such as n!, and powers, such as x^n or 3^n.
- For a power series sum c_n x^n, the Ratio Test often helps find the radius of convergence R.
Vocabulary
- Infinite series
- An infinite series is a sum of infinitely many terms, written as sum a_n.
- Convergence
- Convergence means the partial sums of a series approach a finite number.
- Absolute convergence
- Absolute convergence means the series sum |a_n| converges.
- Ratio Test
- The Ratio Test decides convergence by studying the limit of the absolute value of the ratio of consecutive terms.
- Inconclusive
- Inconclusive means the test does not provide enough information to decide whether the series converges or diverges.
Common Mistakes to Avoid
- Forgetting the absolute value in |a_{n+1}/a_n|. The Ratio Test is based on the size of the ratio, so signs or alternating behavior should not control the limit.
- Saying the series diverges when L = 1. The Ratio Test gives no conclusion in this case, so another test such as the p-series test, comparison test, or alternating series test may be needed.
- Taking the ratio a_n/a_{n+1} instead of a_{n+1}/a_n without adjusting the conclusion. Reversing the ratio changes the limit to a reciprocal, so the L < 1 and L > 1 decisions no longer match.
- Applying the test to the terms but forgetting the final decision is about the series. The fact that terms get smaller is not enough by itself, because convergence depends on the behavior of the infinite sum.
Practice Questions
- 1 Use the Ratio Test to determine whether the series sum from n = 1 to infinity of n!/5^n converges or diverges.
- 2 Use the Ratio Test to determine whether the series sum from n = 1 to infinity of 3^n/n! converges or diverges.
- 3 A student applies the Ratio Test to sum 1/n and gets L = 1. Explain what this result means and name a different test that can decide the series.