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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.8.45
Chapter 10, Problem 10.8.45

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Convergence of an Infinite Series

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.
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Direct Comparison Test

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.
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Exponential Growth & Decay
Related Practice
Textbook Question

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

∑ (k = 1 to ∞) k / eᵏ

Textbook Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.


{(n + 1)!⁄n!}

Textbook Question

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


31.∑ (k = 1 to ∞) 2^(–3k)

Textbook Question

Simplify k! / (k + 2)! for any integer k ≥ 0.

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Textbook Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.


{√((1 + 1 / 2n)ⁿ)}

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Textbook Question

39–44. {Use of Tech} Estimating infinite series Estimate the value of the following convergent series with an absolute error less than 10⁻³.


∑ (k = 1 to ∞) (−1)ᵏ / kᵏ