Problem 18 - Thomas Calculus

Question

In Exercises \(17-46,\) use any method to determine if the series converges or diverges. Give reasons for your answer. $$ \sum_{n=1}^{\infty}(-1)^{n} n^{2} e^{-n} $$

Step-by-Step Solution

Answer

The series converges by the Alternating Series Test.

Step 1: Identify the Type of Series

The series given is \( \sum_{n=1}^{\infty} (-1)^{n} n^{2} e^{-n} \). Notice the \((-1)^{n}\) term, which indicates that this is an alternating series.

Step 2: Apply the Alternating Series Test

The Alternating Series Test requires checking two conditions: 1. The absolute value of the terms \( a_n = n^2 e^{-n} \) must be decreasing.2. The limit of \( a_n \) as \( n \to \infty \) must be zero.Calculate: - \( \lim_{n\to\infty} n^2 e^{-n} = 0 \) because exponential decay is faster than polynomial growth.- The sequence \( n^2 e^{-n} \) is decreasing for large \( n \) because the derivative of \( b_n(n^2 e^{-n}) = (2n-n^2)e^{-n} \) is negative for sufficiently large \( n \).

Step 3: Conclusion of the Alternating Series Test

Since both conditions of the Alternating Series Test are satisfied (terms are decreasing, and the limit is zero), the series \( \sum_{n=1}^{\infty} (-1)^{n} n^2 e^{-n} \) converges.

Key Concepts

Alternating Series Test

An Alternating Series is a series whose terms alternate signs between positive and negative. Such series often have a general form of \[\sum (-1)^n a_n \] where each term \(a_n\) is positive. To determine if an alternating series converges, we use the Alternating Series Test.

This test requires checking two conditions:

• The sequence \(a_n\) must be decreasing.
• The limit \(\lim_{n \to \infty} a_n\) must be zero.

If both conditions are met, the series is guaranteed to converge. This is very useful for problems where terms involve complicated expressions like polynomials and exponentials.

For example, in our series \(\sum_{n=1}^{\infty}(-1)^{n} n^{2} e^{-n}\), we applied the test and found both conditions satisfied, indicating convergence.

Limit of a Sequence

A Limit of a Sequence is a key concept in calculus, where a sequence \(\{a_n\}\) approaches a specific value \(L\) as \(n\) becomes very large, i.e., \(\lim_{n \to \infty} a_n = L\). In the context of our series, determining the limit helps us apply the Alternating Series Test.

Consider the sequence \(a_n = n^2 e^{-n}\). As \(n\) grows, the exponential \(e^{-n}\) decays to zero much more rapidly than the growth of \(n^2\). Consequently, the limit: \[\lim_{n\to\infty} n^2 e^{-n} = 0\]This fast decay to zero is crucial because it ensures that one of the Alternating Series Test conditions is satisfied. Without this, the series would not converge.

Polynomial and Exponential Functions

Polynomial and Exponential Functions are two fundamental types of functions with different growth rates. In our exercise, the term \(n^2 e^{-n}\) combines a polynomial \(n^2\) and an exponential function \(e^{-n}\). Understanding their interaction is essential for evaluating the convergence of the series.

Polynomial functions, like \(n^2\), grow at a relatively slow and consistent rate. On the other hand, exponential functions, such as \(e^{-n}\), decrease very rapidly. When both are present in the same expression, the exponential decay often dominates.

This interaction is why \(n^2 e^{-n}\) tends toward zero as \(n\) increases, leading to the convergence condition in alternating series.

Series and Sequences

Series and Sequences are foundational concepts in analysis. A sequence is essentially an ordered list of numbers, while a series is the sum of the terms of a sequence.

To analyze the convergence of a series, we first examine the behavior of its underlying sequence. For alternating series like ours, we focused on the sequence \(a_n = n^2 e^{-n}\). Determining if this sequence approaches zero helps us apply tests for convergence.

Understanding the connection between sequences and series is crucial, as series depend on the properties of sequences to discern their convergence or divergence. In our exercise, these concepts guide us to accurate conclusions using methods like the Alternating Series Test.