Question

Use the ratio test to determine whether the series \(\sum_{n=1}^{\infty} \frac{n^{3}}{3^{n}}\) converges or diverges.

Short answer

The series converges by the ratio test.

Step-by-step solution

Identify the terms of the series

The given series is \( \sum_{n=1}^{\infty} \frac{n^{3}}{3^{n}} \). We will denote the n-th term of the series as \( a_n = \frac{n^{3}}{3^{n}} \). Our goal is to apply the ratio test to this series.

Apply the Ratio Test

The ratio test involves evaluating the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). Express \( a_{n+1} = \frac{(n+1)^{3}}{3^{n+1}} \) and calculate the ratio \( \frac{a_{n+1}}{a_n} = \frac{(n+1)^{3}}{3^{n+1}} \cdot \frac{3^n}{n^3} \). Simplify the expression: \( \frac{a_{n+1}}{a_n} = \frac{(n+1)^{3}}{3 \cdot n^3} \).

Simplify the ratio

The expression becomes \( \frac{a_{n+1}}{a_n} = \frac{(n+1)^3}{3 \, n^3} \). Expand \( (n+1)^3 = n^3 + 3n^2 + 3n + 1 \), leading to \( \frac{a_{n+1}}{a_n} = \frac{n^3 + 3n^2 + 3n + 1}{3n^3} \). This simplifies to \( \frac{1}{3} + \frac{1}{n} + \frac{1}{n^2} + \frac{1}{3n^3} \).

Evaluate the Limit

Evaluate \( \lim_{n \to \infty} \left( \frac{1}{3} + \frac{1}{n} + \frac{1}{n^2} + \frac{1}{3n^3} \right) \). As \( n \to \infty \), the terms \( \frac{1}{n} \), \( \frac{1}{n^2} \), and \( \frac{1}{3n^3} \) tend to 0. Thus, \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \frac{1}{3} \).

Conclude with the Ratio Test

According to the ratio test, if \( L < 1 \), the series converges. Here, \( L = \frac{1}{3} \) which is less than 1. Therefore, the series \( \sum_{n=1}^{\infty} \frac{n^{3}}{3^{n}} \) converges.

Key concepts

Convergence of Series

Understanding whether a series converges or diverges is an essential part of calculus and analysis. When we talk about the convergence of a series, we’re referring to the behavior of the sum of its terms as the number of terms approaches infinity. The series \(\sum_{n=1}^{\infty} \frac{n^{3}}{3^{n}}\) is a perfect example to explore this concept. If the series converges, it means that its terms get closer and closer to a certain finite value. Conversely, if the series diverges, the sum of its terms either increases indefinitely or oscillates without reaching a limit.To determine convergence, mathematicians often use tests like the ratio test or the root test, each providing different insights into the nature of the series. In this case, we applied the ratio test, which helps us assess whether the series converges by examining the limit of the ratio between consecutive terms.

Limit Evaluation

The process of limit evaluation is crucial when applying the ratio test. It involves checking the behavior of a sequence or series as \(n\) approaches infinity. For the series in question, we defined the term \(a_n = \frac{n^{3}}{3^{n}}\) and the subsequent term \(a_{n+1} = \frac{(n+1)^{3}}{3^{n+1}}\). We need to evaluate \[L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]Calculating this limit involves simplifying the expression:\[\frac{a_{n+1}}{a_n} = \frac{(n+1)^3}{3n^3} = \frac{1}{3} + \frac{1}{n} + \frac{1}{n^2} + \frac{1}{3n^3}\]As \(n\) approaches infinity, the terms \(\frac{1}{n}\), \(\frac{1}{n^2}\), and \(\frac{1}{3n^3}\) approach zero. This simplification leaves us with a result for the limit, \(L = \frac{1}{3}\). The significance of the limit value becomes evident in the next steps when using the ratio test.

Power Series

Power series are an important class of series that often appear in calculus and algebra. They are infinite series of the form \(\sum_{n=0}^{\infty} a_n x^n \), with coefficients \(a_n\) and a variable \(x\). Although the original exercise focuses on ratio test and convergence, understanding power series can enrich your overall grasp of series.In mathematics, power series are used to represent functions in an interval, defining them with infinite polynomials. They play a crucial role in various topics, from solving differential equations to representing functions in complex analysis. While the series from our exercise isn't technically a power series (as it lacks a variable base), examining the structure of series deepens comprehension and shows the interplay of sequences and limits.This isn't the central topic here, but recognizing how series structures can adapt to include variables helps when addressing more complex issues in calculus.

Infinite Series

Infinite series are sequences of numbers where terms are summed endlessly. They extend beyond finite summation, where you handle a limited number of terms, introducing an infinite concept in calculations. The simplest form of an infinite series is exemplified by an arithmetic sequence, but more complex forms like geometric or the series in the exercise also exist. The sum plays a key role, as it often seeks what these terms tend towards when summed indefinitely, or in mathematical terms, converging.In our series \(\sum_{n=1}^{\infty} \frac{n^{3}}{3^{n}} \), recognizing it as an infinite series prompts invoking tests or methods to decide its convergence. Infinite series are foundational in mathematical fields that explore beyond boundaries of finite sums, such as representation of irrational numbers or describing waves and signals in engineering fields.