11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) k⁴ / (eᵏ⁵)

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Identify the given series: \( \sum_{k=1}^{\infty} \frac{k^{4}}{e^{k^{5}}} \). We want to determine if this infinite series converges.

Consider the general behavior of the terms \( a_k = \frac{k^{4}}{e^{k^{5}}} \). Notice that the denominator grows exponentially with respect to \( k^{5} \), while the numerator grows polynomially as \( k^{4} \).

Apply the Comparison Test or Limit Comparison Test by comparing \( a_k \) to a simpler series. Since exponential growth dominates polynomial growth, compare \( a_k \) to \( \frac{1}{e^{k^{5}}} \), which is a convergent series because its terms approach zero very rapidly.

Since \( \sum \frac{1}{e^{k^{5}}} \) converges (it is a series with terms decreasing faster than any geometric series), and \( \frac{k^{4}}{e^{k^{5}}} \leq C \cdot \frac{1}{e^{k^{5}}} \) for some constant \( C \) and sufficiently large \( k \), by the Comparison Test, the original series converges.

Conclude that the series \( \sum_{k=1}^{\infty} \frac{k^{4}}{e^{k^{5}}} \) converges absolutely due to the dominance of the exponential term in the denominator.

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주요 개념

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Convergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to determine whether the series sums to a finite value or diverges to infinity.

Comparison Test

The Comparison Test involves comparing the given series to a known benchmark series. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges. This test helps analyze series with complicated terms.

Exponential vs. Polynomial Growth

Exponential functions grow faster than any polynomial function as the variable approaches infinity. In series terms, if the denominator grows exponentially and the numerator polynomially, the terms tend to zero rapidly, often ensuring convergence.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.∑ (from k = 1 to ∞) k⁴ / (eᵏ⁵)