What is ratio test?

The Ratio Test is a powerful tool in calculus used to determine the convergence or divergence of an infinite series. It's particularly useful when dealing with series involving factorials or exponential terms.

Here's a breakdown of the Ratio Test:

The Test:

Given an infinite series Σ aₙ, where aₙ ≠ 0 for all n, we define the limit:

L = lim (n→∞) |aₙ₊₁ / aₙ|

Based on the value of L, we can draw the following conclusions:

If L < 1: The series Σ aₙ converges absolutely. You can learn more about absolute%20convergence here.
If L > 1 (including L = ∞): The series Σ aₙ diverges.
If L = 1: The test is inconclusive. It provides no information about the convergence or divergence of the series. Another test must be used. Refer to other convergence%20tests.

When to Use It:

The Ratio Test is most effective when:

The series contains factorials (e.g., n!)
The series contains exponential terms (e.g., aⁿ)
The series terms have a relatively simple ratio.

How to Apply It:

Form the Ratio: Calculate the ratio |aₙ₊₁ / aₙ|. This involves replacing 'n' with 'n+1' in the general term aₙ to get aₙ₊₁.
Simplify the Ratio: Simplify the resulting expression as much as possible. Often, terms will cancel out.
Evaluate the Limit: Calculate the limit as n approaches infinity: lim (n→∞) |aₙ₊₁ / aₙ| = L.
Interpret the Result: Use the value of L (L < 1, L > 1, or L = 1) to determine whether the series converges, diverges, or if the test is inconclusive.

Example:

Consider the series Σ (n! / nⁿ).

aₙ = n! / nⁿ
aₙ₊₁ = (n+1)! / (n+1)ⁿ⁺¹
|aₙ₊₁ / aₙ| = |((n+1)! / (n+1)ⁿ⁺¹) / (n! / nⁿ)| = |((n+1)! * nⁿ) / (n! * (n+1)ⁿ⁺¹)| = |(n+1) * nⁿ / (n+1)ⁿ⁺¹| = |nⁿ / (n+1)ⁿ| = |(n / (n+1))ⁿ| = |(1 / (1 + 1/n))ⁿ|
L = lim (n→∞) |(1 / (1 + 1/n))ⁿ| = 1/e (approximately 0.368)

Since L = 1/e < 1, the series converges absolutely. This is a good example to study and understand when looking at the convergence%20of%20series.

Important Note:

If the Ratio Test is inconclusive (L = 1), you'll need to use another test, such as the Root Test, Integral Test, Comparison Test, Limit Comparison Test, or other series%20tests. The Ratio Test doesn't always provide a definitive answer. Understanding the limitations of the ratio test is crucial for sequence%20and%20series problems.