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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 10.1.71a
Chapter 10, Problem 10.1.71a

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


a.The sequence of partial sums for the series1 + 2 + 3 + ⋯ is {1, 3, 6, 10, …}.

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Recall that the sequence of partial sums of a series is formed by adding the terms of the series one by one. For the series 1 + 2 + 3 + ⋯, the nth partial sum is given by \(S_n = 1 + 2 + 3 + \cdots + n\).
Use the formula for the sum of the first n natural numbers: \(S_n = \frac{n(n+1)}{2}\). This formula gives the value of the nth partial sum directly.
Calculate the first few partial sums using the formula: for n=1, \(S_1 = \frac{1 \times 2}{2} = 1\); for n=2, \(S_2 = \frac{2 \times 3}{2} = 3\); for n=3, \(S_3 = \frac{3 \times 4}{2} = 6\); for n=4, \(S_4 = \frac{4 \times 5}{2} = 10\).
Compare these values to the given sequence {1, 3, 6, 10, …}. Since they match exactly, the sequence of partial sums for the series 1 + 2 + 3 + ⋯ is indeed {1, 3, 6, 10, …}.
Therefore, the statement is true because the sequence of partial sums corresponds to the triangular numbers given by the formula \(S_n = \frac{n(n+1)}{2}\).

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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

A sequence of partial sums is formed by adding the terms of a series one by one. For the series 1 + 2 + 3 + ⋯, the nth partial sum is the sum of the first n natural numbers, which creates a new sequence representing cumulative totals.
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Introduction to Sequences

Formula for the Sum of the First n Natural Numbers

The sum of the first n natural numbers is given by the formula n(n + 1)/2. This formula helps quickly find the nth partial sum without adding each term individually, confirming the sequence of partial sums as {1, 3, 6, 10, …}.
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Left, Right, & Midpoint Riemann Sums Example 1

Convergence and Divergence of Series

A series converges if its sequence of partial sums approaches a finite limit; otherwise, it diverges. Since the partial sums 1, 3, 6, 10, … grow without bound, the series 1 + 2 + 3 + ⋯ diverges, meaning it does not sum to a finite value.
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Related Practice
Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


a.Write out the first five terms of the sequence.


Drug elimination

Jack took a 200-mg dose of a pain killer at midnight. Every hour, 5% of the drug is washed out of his bloodstream. Let dₙ be the amount of drug in Jack’s blood n hours after the drug was taken, where d₀ = 200mg.

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Textbook Question

71. Evaluating an infinite series two ways

Evaluate the series

∑ (k = 1 to ∞) (4 / 3ᵏ – 4 / 3ᵏ⁺¹) two ways.

a. Use a telescoping series argument.

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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. Suppose 0 < aₖ < bₖ. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 1 to ∞) bₖ converges.

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Textbook Question

57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.


a. Find the first four terms of the sequence of heights {hₙ}.


h₀ = 20,r = 0.5

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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. A series that converges must converge absolutely.

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Textbook Question

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


a.Write out the first five terms of the sequence.


Radioactive decay

A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.

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