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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.R.65
Chapter 10, Problem 10.R.65

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)k√k / k³

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First, rewrite the general term of the series to simplify it. The term is given as \(\frac{k \sqrt{k}}{k^3}\). Recall that \(\sqrt{k} = k^{1/2}\), so rewrite the term as \(\frac{k \cdot k^{1/2}}{k^3}\).
Combine the powers of \(k\) in the numerator: \(k \cdot k^{1/2} = k^{1 + 1/2} = k^{3/2}\). So the term becomes \(\frac{k^{3/2}}{k^3}\).
Simplify the fraction by subtracting exponents in the denominator from the numerator: \(k^{3/2 - 3} = k^{-3/2}\).
Now, the series can be written as \(\sum_{k=1}^\infty k^{-3/2}\). Recognize this as a p-series of the form \(\sum_{k=1}^\infty \frac{1}{k^p}\) where \(p = \frac{3}{2}\).
Apply the p-series convergence test: a p-series converges if and only if \(p > 1\). Since \(\frac{3}{2} > 1\), conclude that the series converges.

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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 series like the one given.
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Convergence of an Infinite Series

Convergence Tests

Convergence tests are methods used to determine if an infinite series converges or diverges. Common tests include the Comparison Test, Ratio Test, Root Test, and p-Series Test. Choosing an appropriate test depends on the form of the series terms.
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Choosing a Convergence Test

Simplifying Series Terms

Before applying convergence tests, it is important to simplify the general term of the series. For example, rewriting k√k / k³ as k^(1 + 1/2) / k³ = k^(3/2) / k³ = k^(-3/2) helps identify the type of series and apply the correct convergence test.
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Geometric Series
Related Practice
Textbook Question

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = ((3n² + 2n + 1) · sin(n)) / (4n³ + n) (Hint: Use the Squeeze Theorem.)

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Textbook Question

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


g.The series ∑ (from k = 1 to ∞) (k² / (k² + 1)) converges.

Textbook Question

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = (–1)ⁿ / 0.9ⁿ

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Textbook Question

Building a tunnel — first scenario

A crew of workers is constructing a tunnel through a mountain. Understandably, the rate of construction decreases because rocks and earth must be removed a greater distance as the tunnel gets longer. Suppose each week the crew digs 0.95 of the distance it dug the previous week. In the first week, the crew constructed 100 m of tunnel.


a.How far does the crew dig in 10 weeks? 20 weeks? N weeks?

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Textbook Question

Determine whether the following statements are true and give an explanation or counterexample.

c. The terms of the sequence of partial sums of the series ∑ aₖ approach 5/2, so the infinite series converges to 5/2.

Textbook Question

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


e.The sequence aₙ = n² / (n² + 1) converge.

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