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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.1.71a
Capitolo 10, Problema 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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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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Percorso guidato
10:22
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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06:52
Convergence of an Infinite Series
Pratica correlata
Domanda del libro di testo

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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39–40. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, use Theorem 10.13 to complete the following.


a. Use Sₙ to estimate the sum of the series.


39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2

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{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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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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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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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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