Question

Use the ratio test to find whether the following series converge or diverge: $$\sum_{n=0}^{\infty} \frac{3^{n}}{2^{2 n}}$$

Short answer

The series converges because the limit of the ratio test is less than 1.

Step-by-step solution

Identify the General Term

The given series is \ \ \sum_{n=0}^{\infty} \frac{3^{n}}{2^{2n}}. Identify the general term \(a_n\): \ \ \(a_n = \frac{3^n}{2^{2n}} = \frac{3^n}{(2^2)^n} = \left(\frac{3}{4}\right)^n.\)

Set Up the Ratio Test

For the ratio test, calculate \(\lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| \): \ \ \( \frac{a_{n+1}}{a_n} = \frac{\left(\frac{3}{4}\right)^{n+1}}{\left(\frac{3}{4}\right)^n} = \frac{3}{4}. \)

Evaluate the Limit

Evaluate the limit: \ \ \(\lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{3}{4} \right| = \frac{3}{4}. \)

Apply the Ratio Test

According to the ratio test, if \(\lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n} \right| < 1\), the series converges. Since \(\frac{3}{4} < 1\), the series converges.

Key concepts

series convergence

Series convergence is an essential idea in calculus and mathematical analysis. A series is the sum of the terms of a sequence. For example, consider the infinite series \(\textstyle \sum_{n=0}^{\infty} a_n\).
The goal is to determine if this sum approaches a finite value as more and more terms are added. When a series approaches a finite value, we say the series converges.
If it does not approach a finite value, it diverges. For example, in the series \(\textstyle \sum_{n=0}^{\infty} \frac{3^n}{2^{2n}}\), we need to determine if it converges or diverges.
Step-by-step methods like the ratio test give us a systematic way to decide this.
Using such methods helps ensure we reach the correct conclusion about the series' behavior.

ratio test

The ratio test is a useful technique for working with series convergence. To use it, you start by identifying the general term \(a_n\) of your series. For our example, \(\textstyle a_n = \frac{3^n}{2^{2n}} = \frac{3^n}{(2^2)^n} = \left( \frac{3}{4}\right)^n\).
Next, you set up the ratio test by calculating \(\textstyle \frac{a_{n+1}}{a_n}\), which gives you the ratio between consecutive terms. For our series, this is \(\textstyle \frac{ \left( \frac{3}{4}\right)^{n+1}}{ \left( \frac{3}{4}\right)^n} = \frac{3}{4}\).
The key step is evaluating the limit: \(\textstyle \lim_{{n \to \infty}} \left| \frac{a_{n+1}}{a_n}\right|\). If the result is less than 1, then the series converges. If it’s greater than 1, the series diverges.
In our case, \(\textstyle \left| \frac{3}{4} \right| = \frac{3}{4}\ < 1\), so the series converges according to the ratio test.

infinite series

An infinite series is a sum that involves an infinite number of terms. In mathematical notation, it is represented as \(\textstyle \sum_{n=0}^{\infty} a_n\).
The terms \(a_n\) could be any sequence of numbers. An important problem is understanding whether adding up these terms will result in a finite number (converge) or not (diverge).
For example, the series \(\textstyle \sum_{n=0}^{\infty} \frac{3^n}{2^{2n}}\) has an infinite number of terms, and we need to check its convergence.
This is where techniques like the ratio test become incredibly useful. Analyzing infinite series not only helps in theoretical math but has practical applications in physics, engineering, and computer science.
Knowing whether an infinite series converges can help solve real-world problems where infinite processes are modeled mathematically.