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. $$\sum_{n=1}^{\infty} \frac{2^{n} n^{2}}{3^{n}}$$

Short answer

The series converges by the Root Test.

Step-by-step solution

Identify the General Term

The general term of the series is given by \( a_n = \frac{2^{n} n^{2}}{3^{n}} \). We will use this term in the Root Test to determine the convergence of the series.

Apply the Root Test Formula

The Root Test involves finding the \( n \)-th root of the absolute value of \( a_n \). Compute \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \). For this series, calculate \( \lim_{n \to \infty} \sqrt[n]{\frac{2^n n^2}{3^n}} \).

Simplify the Expression

Simplify the expression \( \sqrt[n]{\frac{2^n n^2}{3^n}} = \frac{\sqrt[n]{2^n} \cdot \sqrt[n]{n^2}}{\sqrt[n]{3^n}} \). Note that \( \sqrt[n]{2^n} = 2 \) and \( \sqrt[n]{3^n} = 3 \). Thus, the expression becomes \( \frac{2 \cdot \sqrt[n]{n^2}}{3} \). Further, \( \sqrt[n]{n^2} = n^{2/n} \rightarrow 1 \) as \( n \rightarrow \infty \).

Calculate the Limit

Compute the limit: \( \lim_{n \to \infty} \frac{2 \cdot n^{2/n}}{3} = \frac{2}{3} \).

Conclusion of the Root Test

Since the limit \( \frac{2}{3} < 1 \), by the Root Test, the series \( \sum_{n=1}^{\infty} \frac{2^{n} n^{2}}{3^{n}} \) converges.

Key concepts

Convergence of Series

Understanding the concept of convergence in series is crucial when dealing with mathematical sequences. In essence, a series is a sum of terms of a sequence. When a series converges, the sum of its infinite terms approaches a finite number. Conversely, if the series diverges, the sum tends to infinity or does not settle on a specific number.

There are several tests to determine convergence, but the Root Test is one of the most effective tools. It involves analyzing the limit of the n-th root of the absolute value of terms in a series. If this limit is less than 1, the series converges. If it's greater than 1, the series diverges. If it equals 1, the test is inconclusive, prompting the need for other methods, such as the Ratio Test or the Alternating Series Test, to determine convergence. By using these tests correctly, we can better understand the behavior of infinite series.

Limit of a Sequence

A limit of a sequence describes the value that the terms of a sequence tend towards as the index (often denoted as \( n \)) approaches infinity. This concept is foundational in calculus and analysis, representing the behavior of functions as they extend indefinitely.

For example, consider a sequence defined as \( a_n = \frac{1}{n} \). As \( n \to \infty \), each term decreases, approaching 0. Thus, the limit of this sequence is 0.

In the Root Test, we deal with the limit of the n-th root of the terms in the series. When working through this, we simplify the expression and find the limiting behavior. For the series given in the exercise, simplifying and calculating \( \lim_{n \to \infty} \sqrt[n]{\frac{2^n n^2}{3^n}} \) helps us determine convergence. Limits are powerful in evaluating how series behave in the long run.

Power Series

Power series are series in the form \( \sum_{n=0}^{\infty} a_nx^n \), where each term consists of a constant coefficient \( a_n \) and a variable base \( x \) raised to the power of \( n \). Analyzing power series becomes essential in calculus, complex analysis, and many other mathematical fields.

A key part of working with power series is understanding the region or interval of convergence. This interval shows the values of \( x \) for which the series converges. To find this interval, convergence tests such as the Root Test or Ratio Test are applied.

Power series converge absolutely at every point in their interval of convergence and provide us with useful representations of functions. They are notably valuable in approximations and in solving differential equations. In the context of the given exercise, understanding series convergence is similar, albeit the series is not explicitly a power series. However, the root testing approach still applies in understanding convergence behavior.