Problem 35 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether the series converges or diverges. In this set of problems knowledge of all the convergence tests from the chapter is assumed. \(\sum_{n=1}^{\infty} \frac{n^{3}}{2^{n}}\)

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} \frac{n^{3}}{2^{n}}\) converges.

Step 1: Set up the Ratio Test

The Ratio Test states that for a given series \( \sum_{n=1}^{\infty} a_{n}\), find the limit \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\). If the L < 1, the series converges; if L > 1, it diverges; and if L = 1, the test is inconclusive.

Step 2: Apply the Ratio Test to the given series

calculating \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\) for the given series \( \sum_{n=1}^{\infty} \frac{n^{3}}{2^{n}}\). This gives \(L = \lim_{n \to \infty} \left| \frac{(n+1)^{3}}{2^{n+1}} \times \frac{2^{n}}{n^{3}} \right|\). This simplifies to \(L = \lim_{n \to \infty} \frac{(n+1)^{3}}{2n^{3}}\)

Step 3: Simplify the limit

The term can be simplified to find its limit as \( L = \lim_{n \to \infty} \frac{(n+1)^{3}}{2n^{3}} = \frac{1}{2} \(.

Step 4: Conclude

Since the value of L is less than 1 (L < 1), based on the ratio test, it can be concluded that the given series \( \sum_{n=1}^{\infty} \frac{n^{3}}{2^{n}}\) converges.

Key Concepts

Ratio Test

The Ratio Test is a tool used to determine the convergence or divergence of an infinite series. Using this test, you examine the limit of the absolute value of the ratio of successive terms in a series. For a series \( \sum_{n=1}^{\infty} a_n \), the test requires you to evaluate:\[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \]- If \( L < 1 \), the series converges.- If \( L > 1 \) or is infinite, the series diverges.- If \( L = 1 \), the test is inconclusive.This test is particularly useful for series with factorials or exponential factors, as these components simplify well within the test's framework.

limit evaluation

Evaluating limits is an essential step when using the Ratio Test. In this context, once the expression \( \left| \frac{a_{n+1}}{a_n} \right| \) is set up for the series, it often needs simplification to a form where the limit can be directly computed.For example, consider the expression from our test for the series:\[ \lim_{n \to \infty} \left| \frac{(n+1)^3}{2^{n+1}} \times \frac{2^n}{n^3} \right| \]This can be rearranged and simplified to:\[ \lim_{n \to \infty} \frac{(n+1)^3}{2n^3} \]By expanding \((n+1)^3 = n^3 + 3n^2 + 3n + 1\), and applying polynomial division, this simplifies down further to \( \frac{1}{2} \). Recognizing when and how to simplify these terms is pivotal for finding the limit accurately.

convergent series

A convergent series is an infinite series that adds up to a finite number. In simpler terms, as you add more and more terms of the series, the total sum approaches a specific finite value.When using tests like the Ratio Test, determining \( L < 1 \) indicates convergence. This means that the series behaves in such a way that the later terms become very small and insignificant compared to the earlier ones.Convergent series are crucial in various mathematical and applied contexts because they assure us that a series does not lead to infinity or undefined behaviors, making it possible to use them in calculations and theoretical analysis.

divergent series

Divergent series are those that do not converge to a finite limit as more terms are added. If a series diverges, its terms do not settle down to add up to a specific value, and this makes it quite unpredictable.In the context of the Ratio Test, if \( L > 1 \) or if the test results in \( L = \infty \), then the series is divergent. This essentially means that subsequent terms are not reducing to zero adequately fast enough, causing the series to grow indefinitely.Divergent series, when encountered, often require careful consideration. Sometimes, they can be altered or rearranged under certain conditions to form convergent ones, but typically, they signal the need for different approaches or interpretations in mathematical calculations.