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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.1c
Chapter 10, Problem 10.R.1c

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.

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Recall that the sequence of partial sums \( S_n = \sum_{k=1}^n a_k \) represents the sum of the first \( n \) terms of the series \( \sum a_k \).
If the terms of the sequence of partial sums \( S_n \) approach a limit \( L \) as \( n \to \infty \), then the infinite series \( \sum a_k \) converges to \( L \).
In this problem, it is given that the terms of the sequence of partial sums approach \( \frac{5}{2} \). This means \( \lim_{n \to \infty} S_n = \frac{5}{2} \).
Since the partial sums approach \( \frac{5}{2} \), by definition, the infinite series \( \sum a_k \) converges and its sum is \( \frac{5}{2} \).
Therefore, the statement is true because the convergence of the partial sums to \( \frac{5}{2} \) directly implies the series converges to \( \frac{5}{2} \).

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Key Concepts

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

Sequence of Partial Sums

The sequence of partial sums is formed by adding the first n terms of a series. It helps analyze the behavior of the series by examining the limit of these sums as n approaches infinity.
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Convergence of an Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. This limit is the sum of the series, meaning the series adds up to a specific value.
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Convergence of an Infinite Series

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence get arbitrarily close to as the index grows large. If the partial sums approach 5/2, the series converges to 5/2.
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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

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

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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Textbook Question

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

Textbook Question

89–90. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, complete the following.


c. Find lower and upper bounds (Lₙ and Uₙ respectively) for the exact value of the series.


89.∑ (from k = 1 to ∞)1 / k⁵ ;n = 5