Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 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, …}.

Verified step by step guidance

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.

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, …}.

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

{Use of Tech} A savings plan

James begins a savings plan in which he deposits \(100 at the beginning of each month into an account that earns 9% interest annually, or equivalently, 0.75% per month.

To be clear, on the first day of each month, the bank adds 0.75% of the current balance as interest, and then James deposits \)100.

Let Bₙ be the balance in the account after the nᵗʰ payment, where B₀ = \$0.

a.Write the first five terms of the sequence {Bₙ}.

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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 = 30,r = 0.25

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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.

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