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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 10.3.39
Chapter 10, Problem 10.3.39

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


39.∑ (k = 2 to ∞) (–0.15)ᵏ

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Identify the first term of the geometric series by substituting the lower limit of the summation into the general term: the first term is \(a = (-0.15)^2\).
Determine the common ratio \(r\) of the geometric series, which is the base of the exponent in the term, so \(r = -0.15\).
Check the convergence of the series by evaluating the absolute value of the common ratio: if \(|r| < 1\), the series converges; otherwise, it diverges.
Since the series starts at \(k=2\) instead of \(k=0\) or \(k=1\), express the sum from \(k=2\) to infinity in terms of the sum from \(k=0\) to infinity by factoring out the first two terms or adjusting the formula accordingly.
Use the formula for the sum of an infinite geometric series \(S = \frac{a}{1 - r}\), where \(a\) is the first term of the series starting at \(k=2\), to write the sum expression without calculating the final numerical value.

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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. Understanding the structure helps determine convergence and sum.
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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. If |r| ≥ 1, the series diverges. This criterion is essential to decide whether the series has a finite sum or not.
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Convergence of an Infinite Series

Sum Formula for Convergent Geometric Series

When an infinite geometric series converges, its sum is given by S = a / (1 - r), where a is the first term and r is the common ratio. This formula allows direct calculation of the series sum once convergence is established.
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Related Practice
Textbook Question

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

{Use of Tech} ∑ (k = 1 to ∞) 1 / (4lnk)

Textbook Question

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.


∑ (from k = 1 to ∞)1 / ln(eᵏ + 1)

Textbook Question

Is it possible for a series of positive terms to converge conditionally? Explain.

Textbook Question

9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

∑ (from k = 1 to ∞) (2ᵏ / k⁹⁹)

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 = 3 + cos(π*ⁿ) ; n = 1, 2, 3, …

Textbook Question

Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.


{nsin(6 / n)}

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