Problem 18 - Thomas Calculus

Question

Determining Convergence or Divergence In Exercises \(17-44,\) 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: Recognize the Series Type

The given series is \(\sum_{n=1}^{\infty}(-1)^{n} n^{2} e^{-n}\). This is an alternating series because it has the term \((-1)^{n}\), which alternates between positive and negative.

Step 2: Check for Alternating Series Test

The Alternating Series Test states that if the absolute value sequence \(b_n = n^2 e^{-n}\) is decreasing and \(\lim_{n \to \infty} b_n = 0\), then the alternating series converges.

Step 3: Analyze the Sequence \(b_n\)

Consider \(b_n = n^2 e^{-n}\). As \(n\) increases, \(e^{-n}\) decreases very rapidly to 0, faster than the quadratic term \(n^2\) increases. Thus, \(b_n\) ultimately decreases to 0.

Step 4: Determine Limit of \(b_n\)

Evaluate \(\lim_{n \to \infty} n^2 e^{-n}\). The exponential \(e^{-n}\) decreases towards 0 much faster than \(n^2\) increases, so \(\lim_{n \to \infty} n^2 e^{-n} = 0\).

Step 5: Conclusion Using Alternating Series Test

Since \(b_n = n^2 e^{-n}\) is positive, decreasing, and approaches 0, the series \(\sum_{n=1}^{\infty}(-1)^{n} n^{2} e^{-n}\) converges by the Alternating Series Test.

Key Concepts

Alternating Series

An alternating series is a type of mathematical series where the signs of its terms alternate between positive and negative as the series progresses. This can be recognized by the inclusion of a factor such as \((-1)^n\) in the general term of the series. The presence of \((-1)^n\) indicates that each successive term in the sequence will switch its sign, generating an alternating pattern.
Such series can have different forms but are unified by this alternating feature. Examples include expressions like \((-1)^n a_n\) or \((-1)^{n+1} a_n\), where \(a_n\) is a sequence with positive terms. Alternating series are important in mathematical analysis because they present unique properties for convergence.
In many cases, these series can sum to a finite value even if they stretch to infinity, a quality explored deeper through convergence tests.

Alternating Series Test

The Alternating Series Test is a common method used to determine whether an alternating series converges. It works on evaluating the terms of the sequence based on specific criteria. Here’s how the test applies:

• The terms \((b_n)\) of the series must all be positive.
• The sequence \((b_n)\) should be decreasing, which means each term is less than or equal to the previous one, starting from some point.
• The limit of \((b_n)\) as \(n\) approaches infinity must be zero, which is expressed mathematically as \(\lim_{n \to \infty} b_n = 0\).

If these conditions are met, the Alternating Series Test confirms that the series converges. For example, with \((-1)^n n^2 e^{-n}\), applying the test shows that because \(n^2 e^{-n}\) decreases and tends to zero, the series converges.

Limit of a Sequence

Understanding the limit of a sequence is crucial in many areas of mathematics, especially when analyzing series convergence. The limit of a sequence \(a_n\) as \(n\) approaches infinity is the value that the terms of \(a_n\) approach.
Mathematically, it is written as \(\lim_{n \to \infty} a_n\). If this limit exists and is finite, the sequence is considered convergent. If the limit does not exist or is infinite, the sequence diverges.
In the context of the exercise, determining \ \lim_{n \to \infty} n^2 e^{-n} = 0 \ is vital. Although \(n^2\) grows with \(n,\) the factor \(e^{-n}\) decreases much faster, driving the product to zero. Understanding this balance of exponential decay versus polynomial growth is key to several mathematical analyses.