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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.R.3
Chapter 10, Problem 10.R.3

Geometric sums
Evaluate the geometric sums
∑ (from k = 0 to 9) (0.2)ᵏand∑ (from k = 2 to 9) (0.2)ᵏ.

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1
Recognize that both sums are geometric series where each term is of the form \(r^k\) with common ratio \(r = 0.2\).
Recall the formula for the sum of the first \(n+1\) terms of a geometric series starting at \(k=0\): \(S = \frac{1 - r^{n+1}}{1 - r}\).
For the first sum \(\sum_{k=0}^{9} (0.2)^k\), identify \(n=9\) and apply the formula: \(S_1 = \frac{1 - (0.2)^{10}}{1 - 0.2}\).
For the second sum \(\sum_{k=2}^{9} (0.2)^k\), express it as the difference between the sum from \(k=0\) to \(9\) and the sum from \(k=0\) to \(1\): \(S_2 = \sum_{k=0}^{9} (0.2)^k - \sum_{k=0}^{1} (0.2)^k\).
Calculate \(\sum_{k=0}^{1} (0.2)^k\) using the geometric sum formula with \(n=1\): \(S_{0\text{ to }1} = \frac{1 - (0.2)^2}{1 - 0.2}\), then subtract this from \(S_1\) to find \(S_2\).

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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 terms where each term is found by multiplying the previous term by a constant ratio. It has the form ∑ ar^k, where a is the first term and r is the common ratio. Understanding this structure is essential for evaluating sums like those given.
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Geometric Series

Formula for the Sum of a Finite Geometric Series

The sum of the first n+1 terms of a geometric series is given by S = a(1 - r^(n+1)) / (1 - r), where a is the first term and r is the common ratio (r ≠ 1). This formula allows quick calculation of sums without adding each term individually.
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Index Shifting in Summations

When the summation index does not start at zero, it is often helpful to rewrite the sum by shifting the index to start at zero. This simplifies applying the geometric series formula by adjusting the first term and the number of terms accordingly.
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Related Practice
Textbook Question

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞)k⁴ / √(9k¹² + 2)

Textbook Question

27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.

∑ (from k = 0 to ∞)(tan⁻¹(k + 2) − tan⁻¹k)

Textbook Question

77–87. Absolute or conditional convergence

Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞)(−2)ᵏ⁺¹ / k²

Textbook Question

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 1 to ∞)3 / (2 + eᵏ)

Textbook Question

25–26. Recursively defined sequences

The following sequences {aₙ} from n = 0 to ∞ are defined by a recurrence relation. Assume each sequence is monotonic and bounded.


a.Find the first five terms a₀, a₁, ..., a₄ of each sequence.


25.aₙ₊₁ = (1 / 2) aₙ + 8;a₀ = 80

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Textbook Question

42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.

∑ (from k = 3 to ∞)ln(k) / k³ᐟ²