Skip to main content
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 10.8.35
Chapter 10, Problem 10.8.35

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 2⁹k / kᵏ

Verified step by step guidance
1
First, write down the general term of the series: \(a_k = \frac{2^{9k}}{k^k}\).
To determine convergence, consider applying the Root Test, which is useful for series with terms raised to the power of \(k\). The Root Test uses the limit \(L = \lim_{k \to \infty} \sqrt[k]{|a_k|}\).
Calculate \(\sqrt[k]{|a_k|} = \sqrt[k]{\frac{2^{9k}}{k^k}} = \frac{2^9}{k}\), since \(\sqrt[k]{2^{9k}} = 2^9\) and \(\sqrt[k]{k^k} = k\).
Evaluate the limit \(L = \lim_{k \to \infty} \frac{2^9}{k}\). As \(k\) approaches infinity, \(\frac{2^9}{k}\) approaches 0.
Since \(L = 0 < 1\), by the Root Test, the series \(\sum_{k=1}^\infty \frac{2^{9k}}{k^k}\) converges absolutely.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3m
Was this helpful?

Key Concepts

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

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining whether a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to analyze the behavior of the given series.
Recommended video:
06:52
Convergence of an Infinite Series

Root Test for Convergence

The Root Test involves taking the k-th root of the absolute value of the k-th term and examining its limit as k approaches infinity. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges. This test is particularly useful for series with terms raised to the k-th power.
Recommended video:
07:15
Root Test

Exponential and Factorial Growth Rates

Comparing growth rates of functions like exponentials and powers is crucial in convergence tests. In the series ∑ (2^(9k) / k^k), the denominator grows faster than any exponential due to k^k, which tends to infinity much faster, suggesting the terms approach zero rapidly, influencing convergence.
Recommended video:
5:22
Factorials
Related Practice
Textbook Question

43–44. Periodic doses

Suppose you take a dose of m mg of a particular medication once per day. Assume f equals the fraction of the medication that remains in your blood one day later. Just after taking another dose of medication on the second day, the amount of medication in your blood equals the sum of the second dose and the fraction of the first dose remaining in your blood, which is m + mf. Continuing in this fashion, the amount of medication in your blood just after your nth dose is


Aₙ = m + mf + ⋯ + mfⁿ⁻¹.


For the given values of f and m, calculate A₅, A₁₀, A₃₀, and lim (n → ∞) Aₙ. Interpret the meaning of the limit lim (n → ∞) Aₙ.


43.f = 0.25,m = 200 mg

Textbook Question

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

∑ (from k = 1 to ∞) ((-7)ᵏ / k²)

Textbook Question

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

∑ (from k = 1 to ∞) ((-1)ᵏ) / (k!)

Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.


∑ (from k = 1 to ∞)(1 + 1 / (2k))ᵏ

Textbook Question

38–39. Examining a series two ways Determine whether the following series converge using either the Comparison Test or the Limit Comparison Test. Then use another method to check your answer.

39. ∑ (k = 1 to ∞) 1 / (k² + 2k + 1)

Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞)tan⁻¹(1 / √k)