Question

Determine the convergence of the given series using the Root Test. If the Root Test is inconclusive, state so and determine convergence with another test. $$\sum_{n=1}^{\infty} \frac{3^{n}}{n^{2} 2^{n+1}}$$

Short answer

The series \( \sum_{n=1}^{\infty} \frac{3^{n}}{n^{2} 2^{n+1}} \) diverges by the Root Test.

Step-by-step solution

Identify the series

First, let's identify that we are working with the series \( \sum_{n=1}^{\infty} \frac{3^{n}}{n^{2} 2^{n+1}} \). We need to test this series for convergence.

Apply the Root Test

The Root Test involves calculating \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \) where \( a_n = \frac{3^n}{n^2 2^{n+1}} \). Simplifying, we get \( |a_n| = \frac{3^{n}}{n^{2} 2^{n+1}} = \frac{3^{n}}{2 \cdot 2^{n} n^{2}} = \frac{(\frac{3}{2})^n}{n^2} \). Now calculate \( \sqrt[n]{|a_n|} = \sqrt[n]{\frac{(\frac{3}{2})^n}{n^2}} = \frac{\frac{3}{2}}{\sqrt[n]{n^2}} \).

Evaluate the limit

Now calculate the limit: \( \lim_{n \to \infty} \frac{\frac{3}{2}}{\sqrt[n]{n^2}} \). As \( n \to \infty \), \( \sqrt[n]{n^2} \to 1 \) because \( n^{1/n} \to 1 \). Thus, we have \( \lim_{n \to \infty} \frac{\frac{3}{2}}{1} = \frac{3}{2} \).

Analyze the Root Test result

According to the Root Test, if \( L < 1 \), the series converges, if \( L > 1 \), the series diverges, and if \( L = 1 \), the test is inconclusive. Since \( L = \frac{3}{2} > 1 \), the series diverges.

Key concepts

Convergence Tests

Convergence tests are a toolbox of methods that help us determine whether a given infinite series converges or diverges. These tests are essential because convergence means the infinite series adds up to a finite value, while divergence indicates it does not. Different convergence tests work better with different types of series, and choosing the correct test is often key to solving the problem.

Some common convergence tests include:

• The Root Test: Used primarily when the series has terms involving powers of some expression. It evaluates the n-th root of the absolute value of the n-th term.
• The Ratio Test: Helpful for series with factorials or exponentials. It looks at the limit of the ratio of successive terms.
• The Comparison Test: Compares the series with another series of known convergence.
• The Integral Test: Associates a function with the series and uses integration to determine convergence.

Each test has criteria: if the condition is met, convergence or divergence is confirmed. However, some tests, like the Root Test, may be inconclusive for certain series values, necessitating additional tests for a final verdict.

Infinite Series

An infinite series is a sum of infinitely many terms. It takes the form \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) denotes each term in the series. This structure implies that, unlike finite sums, infinite series do not have a terminal point; they extend indefinitely.

Infinite series are pivotal in many areas of mathematics, such as calculus and analysis. They help model phenomena like population growth and wave patterns. Finite sums easily compute a specific value, but with infinite series, the concept of convergence is crucial. Some infinite series converge to a finite value, despite having infinite terms, while others diverge, indicating growth without bounds or fluctuation without settling.

When considering an infinite series, evaluating its convergence involves exploring if the summation of its infinite terms will settle towards a comfortable, finite number, or if the series will continue without bound. This highlights the significance of understanding and applying convergence tests to series.

Divergent Series

Divergent series are those infinite series that do not converge to a finite sum. As terms are added indefinitely, the series does not approach a specific fixed number. Instead, the sum might increase without bound, oscillate, or behave erratically as more terms are included.

For example, a simple series like \( 1 + 1 + 1 + ... \) continues to grow larger without ever reaching a stable, finite sum, clearly illustrating divergence. In contrast to convergent series, divergent series are often less useful in providing steady results for practical problems.

Different tests, such as the Root Test, can reveal whether a series is divergent. As seen in the original problem, the Root Test determined the series diverged because the calculated limit was greater than 1. Identifying divergence is crucial, as it helps us understand the limitations of using certain series in calculations and ensures that solutions using series remain valid.