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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 10.3.25
Chapter 10, Problem 10.3.25

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


25.∑ (k = 0 to ∞) 0.9ᵏ

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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 \(r\). Here, \(a = 0.9^0 = 1\) and \(r = 0.9\).
Check the convergence condition for a geometric series. A geometric series converges if and only if \(|r| < 1\).
Since \(|0.9| < 1\), the series converges. Use the formula for the sum of an infinite geometric series: \(S = \frac{a}{1 - r}\).
Substitute the values of \(a\) and \(r\) into the formula to express the sum: \(S = \frac{1}{1 - 0.9}\). This expression represents the sum of the 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 quick evaluation of the series without summing infinitely many terms.
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Related Practice
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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Textbook Question

Does a geometric series always have a finite value?

Textbook Question

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 1 to ∞) (−1)ᵏ (k¹¹ + 2k⁵ + 1) / [4k(k¹⁰ + 1)]

Textbook Question

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

∑ (k = 1 to ∞) 1 / ∛k

Textbook Question

55–70. More sequences

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


{nsin³(nπ / 2) / (n + 1)}"

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