Question

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

Accepted Answer

Identify the given series: $$ \sum_{k=1}^{\infty} (-1)^k k e^{-k} . $$This is an alternating series because of the factor $$ (-1)^k . To $$check for absolute convergence, consider the absolute value of the terms: $$ \sum_{k=1}^{\infty} \left| (-1)^k k e^{-k} ight| = \sum_{k=1}^{\infty} k e^{-k} . $$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. Apply the Ratio Test to $$ a_k = k e^{-k} $$: compute $$ \lim_{k 	o \infty} \left| \frac{a_{k+1}}{a_k} ight| = \lim_{k 	o \infty} \frac{(k+1) e^{-(k+1)}}{k e^{-k}} $$ and simplify the expression to determine if the limit is less than 1. 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 $$ k 	o \infty .$$

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 video answer for a similar problem:

Key Concepts

Absolute Convergence

Conditional Convergence

Tests for Convergence of Series

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