Determining Convergence or Divergence
In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑(from n=1 to ∞) [(-1)ⁿ n² e⁻ⁿ]

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Identify the given series: \( \sum_{n=1}^{\infty} (-1)^n n^2 e^{-n} \). This is an alternating series because of the factor \( (-1)^n \), which causes the terms to alternate in sign.

Check the absolute value of the terms: \( a_n = n^2 e^{-n} \). Since \( e^{-n} = \frac{1}{e^n} \), the terms involve a polynomial factor \( n^2 \) multiplied by an exponential decay \( e^{-n} \).

Determine if the terms \( a_n = n^2 e^{-n} \) approach zero as \( n \to \infty \). Because exponential decay dominates polynomial growth, \( a_n \to 0 \). This is a necessary condition for convergence.

Consider the absolute convergence by examining the series \( \sum_{n=1}^{\infty} n^2 e^{-n} \). Since \( e^{-n} \) decays faster than any polynomial grows, this series converges by comparison to a convergent geometric series.

Conclude that since the series converges absolutely (the series of absolute values converges), the original alternating series \( \sum_{n=1}^{\infty} (-1)^n n^2 e^{-n} \) converges as well.

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주요 개념

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Convergence and Divergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely; otherwise, it diverges. Understanding this concept is fundamental to analyzing whether a given series sums to a finite value or not.

Alternating Series Test

The Alternating Series Test determines convergence for series whose terms alternate in sign. If the absolute value of the terms decreases monotonically to zero, the series converges. This test is useful for series with factors like (-1)^n.

Comparison and Limit Comparison Tests

These tests compare a given series to a known benchmark series to determine convergence or divergence. By analyzing the behavior of terms relative to simpler series, one can infer the original series' behavior, especially when exponential or polynomial terms are involved.

Determining Convergence or DivergenceIn Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.∑(from n=1 to ∞) [(-1)ⁿ n² e⁻ⁿ]