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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 10.1.63
Chapter 10, Problem 10.1.63

61–66. Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.


4 + 0.9 + 0.09 + 0.009 + ⋯  

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1
Identify the given infinite series: 4 + 0.9 + 0.09 + 0.009 + \(\cdots\). Notice that each term after the first is obtained by multiplying the previous term by 0.1, so this is a geometric series.
Write the first four terms explicitly: \(a_1 = 4\), \(a_2 = 0.9\), \(a_3 = 0.09\), and \(a_4 = 0.009\).
Calculate the first four partial sums \(S_n\), where \(S_n = a_1 + a_2 + \cdots + a_n\). So, \(S_1 = 4\), \(S_2 = 4 + 0.9\), \(S_3 = 4 + 0.9 + 0.09\), and \(S_4 = 4 + 0.9 + 0.09 + 0.009\).
Recognize that since this is a geometric series with first term \(a = 4\) and common ratio \(r = 0.1\), the sum of the first \(n\) terms can be expressed as \(S_n = a \frac{1 - r^n}{1 - r}\).
To conjecture the value of the infinite series, consider the limit of \(S_n\) as \(n \to \infty\). Since \(|r| < 1\), the infinite sum converges to \(S = \frac{a}{1 - r}\).

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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 an infinite series, the nth partial sum is the sum of the first n terms. Studying these sums helps determine whether the series converges to a finite value or diverges.
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Introduction to Sequences

Geometric Series

A geometric series is a series where each term is found by multiplying the previous term by a constant ratio. Its partial sums have a specific formula, and if the absolute value of the ratio is less than one, the series converges to a finite limit.
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Convergence and Divergence of Series

A series converges if its sequence of partial sums approaches a finite limit as the number of terms grows. If the partial sums grow without bound or oscillate indefinitely, the series diverges. Determining convergence is key to understanding the behavior of infinite series.
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Related Practice
Textbook Question

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 2 to ∞) (5lnk) / k

Textbook Question

9–15. Geometric sums Evaluate each geometric sum.


{Use of Tech}∑ k = 0 to 9(−3/4)ᵏ

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Textbook Question

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 1 to ∞) tan(1 / k)

Textbook Question

45–48. {Use of Tech} Explicit formulas for sequences Consider the formulas for the following sequences {aₙ}ₙ₌₁∞

 Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.

aₙ = ⁿ² + n ;n = 1, 2, 3, …

2
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Textbook Question

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞) 2ᵏ k! / kᵏ

Textbook Question

54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


65. ∑ (k = 1 to ∞) (1 / √(k + 1) – 1 / √(k + 3))