📘 Chapter 16: D’Alembert’s Ratio Test

🧮 Test Statement

Given a series:

\[ \sum_{n=1}^{\infty} a_n \]

Define the limit:

\[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]

• If \( L < 1 \), the series converges absolutely.
• If \( L > 1 \) or \( L = \infty \), the series diverges.
• If \( L = 1 \), the test is inconclusive.

🧠 Intuition Behind It

This test compares the given series to a geometric series. If the terms decrease fast enough (like a convergent geometric progression), then the series also converges. If the ratio between terms grows or doesn’t decrease sufficiently, divergence or indeterminacy can occur.

📌 Example

Consider the series:

\[ \sum_{n=1}^{\infty} \frac{1}{n!} \]

Let \[ ( a_n = \frac{1}{n!}) \]

Then, apply the ratio test:

Now take the limit:

\[ \lim_{n \to \infty} \frac{1}{n+1} = 0 \]

Since \[ ( 0 < 1 ), \] the series converges absolutely.

📝 Practice

Try it yourself: Use the Ratio Test to determine if the following series converges:

\[ \sum_{n=1}^{\infty} \frac{n}{2^n} \]