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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 10.3.21
Chapter 10, Problem 10.3.21

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


21.∑ (k = 0 to ∞) (1/4)ᵏ

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Identify the type of series given. The series is a geometric series of the form \(\sum_{k=0}^{\infty} ar^k\), where \(a\) is the first term and \(r\) is the common ratio.
Determine the first term \(a\) and the common ratio \(r\) from the series \(\sum_{k=0}^{\infty} \left(\frac{1}{4}\right)^k\). Here, \(a = 1\) (since when \(k=0\), the term is \(\left(\frac{1}{4}\right)^0 = 1\)) and \(r = \frac{1}{4}\).
Check the convergence of the series by examining the absolute value of the common ratio \(r\). The geometric series converges if and only if \(|r| < 1\). Since \(|\frac{1}{4}| = \frac{1}{4} < 1\), the series converges.
Use the formula for the sum of an infinite geometric series that converges: \(S = \frac{a}{1 - r}\). Substitute \(a = 1\) and \(r = \frac{1}{4}\) into this formula.
Write the expression for the sum of the series as \(S = \frac{1}{1 - \frac{1}{4}}\). This expression represents the sum of the infinite geometric series.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Geometric Series

A geometric series is the sum of the terms of a geometric sequence, where each term is found by multiplying the previous term by a constant ratio. It is expressed as ∑ ar^k, where a is the first term and r is the common ratio.
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Geometric Series

Convergence of Infinite Geometric Series

An infinite geometric series converges if the absolute value of the common ratio |r| is less than 1. When it converges, the sum can be calculated using the formula S = a / (1 - r). If |r| ≥ 1, the series diverges.
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Convergence of an Infinite Series

Sum Formula for Infinite Geometric Series

For a convergent infinite geometric series with first term a and common ratio r (|r| < 1), the sum is given by S = a / (1 - r). This formula allows direct computation of the series sum without adding infinitely many terms.
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Related Practice
Textbook Question

Define the remainder of an infinite series.

Textbook Question

35–44. Limits of sequences Write the terms a₁, a₂, a₃, and a₄ of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. 

aₙ = 1⁄10ⁿ; n = 1, 2, 3, …

Textbook Question

1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.

∑ (from k = 3 to ∞) (2k²) / (k² − k − 2)

Textbook Question

6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.


{(−0.7)ⁿ}

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Textbook Question

For what values of r does the sequence {rⁿ} converge? Diverge?

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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 ∞) (−1)ᵏ k³ / √(k⁸ + 1)