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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 10.3.87a
Chapter 10, Problem 10.3.87a

87. Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.


a. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 10 to ∞) aₖ converges.

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1
Understand the problem: We are asked to determine if the convergence of the infinite series \(\sum_{k=1}^{\infty} a_k\) implies the convergence of the series \(\sum_{k=10}^{\infty} a_k\).
Recall the definition of convergence for infinite series: A series \(\sum_{k=m}^{\infty} a_k\) converges if the sequence of partial sums \(S_n = \sum_{k=m}^{n} a_k\) approaches a finite limit as \(n \to \infty\).
Consider the relationship between the two series: The series starting at \(k=10\) is essentially the tail of the series starting at \(k=1\). Specifically, \(\sum_{k=1}^{\infty} a_k = \sum_{k=1}^{9} a_k + \sum_{k=10}^{\infty} a_k\).
Since the sum of the first 9 terms, \(\sum_{k=1}^{9} a_k\), is a finite number, subtracting it from the convergent series \(\sum_{k=1}^{\infty} a_k\) leaves the tail \(\sum_{k=10}^{\infty} a_k\), which must also converge.
Therefore, the convergence of \(\sum_{k=1}^{\infty} a_k\) guarantees the convergence of \(\sum_{k=10}^{\infty} a_k\) because removing a finite number of terms from the start of a convergent series does not affect its convergence.

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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. A series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether changing the starting index affects the sum's behavior.
Recommended video:
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Convergence of an Infinite Series

Effect of Finite Number of Terms on Convergence

Adding or removing a finite number of terms from an infinite series does not affect its convergence. Since convergence depends on the tail behavior of the series, starting the sum at k=10 instead of k=1 preserves convergence if the original series converges.
Recommended video:
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Divergence Test (nth Term Test)

Partial Sums and Tail of a Series

Partial sums are sums of the first n terms of a series. The tail of a series refers to the sum from some index onward. Convergence depends on the tail's limit, so analyzing the tail from k=10 onward helps determine if the series remains convergent.
Recommended video:
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Intro to Series: Partial Sums
Related Practice
Textbook Question

Explain why or why not

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

a.If limₙ→∞aₙ = 1 and limₙ→∞bₙ = 3, then limₙ→∞(bₙ / aₙ) = 3.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. The sum ∑ (k = 1 to ∞) 1 / 3ᵏ is a p-series.

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Textbook Question

{Use of Tech} Drug Dosing

A patient takes 75 mg of a medication every 12 hours; 60% of the medication in the blood is eliminated every 12 hours.



a.Let dₙ equal the amount of medication (in mg) in the bloodstream after n doses, where d₁ = 75.

Find a recurrence relation for dₙ.

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Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

a. Find the next two terms of the sequence.


{1, 3, 9, 27, 81, ......}

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. n!n! = (2n)! for all positive integers n.

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Textbook Question

{Use of Tech} Repeated square roots

Consider the sequence defined by

aₙ₊₁ = √(2 + aₙ),a₀ = √2, for n = 0, 1, 2, 3, …


a.Evaluate the first four terms of the sequence {aₙ}.

State the exact values first, and then the approximate values.

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