77–87. Absolute or conditional convergence
Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞)(−1)ᵏk·e⁻ᵏ

Verified step by step guidance

1. Identify the given series: \( \sum_{k=1}^{\infty} (-1)^k k e^{-k} \). This is an alternating series because of the factor \( (-1)^k \).
2. To check for absolute convergence, consider the absolute value of the terms: \( \sum_{k=1}^{\infty} \left| (-1)^k k e^{-k} \right| = \sum_{k=1}^{\infty} k e^{-k} \).
3. Analyze the absolute value series \( \sum_{k=1}^{\infty} k e^{-k} \). Since \( e^{-k} = \frac{1}{e^k} \), the terms look like \( \frac{k}{e^k} \). Use a convergence test suitable for series with terms involving \( k \) and exponential decay, such as the Ratio Test.
4. Apply the Ratio Test to \( a_k = k e^{-k} \): compute \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = \lim_{k \to \infty} \frac{(k+1) e^{-(k+1)}}{k e^{-k}} \) and simplify the expression to determine if the limit is less than 1.
5. If the absolute value series converges, then the original series converges absolutely. If it does not, check if the original alternating series converges by applying the Alternating Series Test, which requires that \( k e^{-k} \) decreases to zero as \( k \to \infty \).

Key Concepts

Absolute Convergence

A series ∑a_k converges absolutely if the series of absolute values ∑|a_k| converges. Absolute convergence guarantees convergence regardless of the sign of terms, and it implies the original series converges. Testing absolute convergence often involves comparison or ratio tests.

Conditional Convergence

A series converges conditionally if it converges, but does not converge absolutely. This means ∑a_k converges, but ∑|a_k| diverges. Conditional convergence often occurs in alternating series where the terms decrease in magnitude and approach zero.

Tests for Convergence of Series

To determine convergence, tests like the Ratio Test, Root Test, and Alternating Series Test are used. The Ratio Test is useful for series with exponential terms, while the Alternating Series Test applies to series with alternating signs and decreasing terms. These tests help classify the series as absolutely convergent, conditionally convergent, or divergent.