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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 10.3.35
Chapter 10, Problem 10.3.35

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


35.∑ (k = 0 to ∞) 3(–π)^(–k)

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Identify the first term \( a \) of the geometric series by substituting \( k = 0 \) into the general term \( 3(-\pi)^{-k} \). This gives \( a = 3(-\pi)^0 = 3 \).
Determine the common ratio \( r \) by finding the factor that each term is multiplied by to get the next term. This is \( r = \frac{3(-\pi)^{-1}}{3(-\pi)^0} = (-\pi)^{-1} = \frac{1}{-\pi} = -\frac{1}{\pi} \).
Check the convergence of the series by evaluating the absolute value of the common ratio \( |r| = \left| -\frac{1}{\pi} \right| = \frac{1}{\pi} \). Since \( \pi > 1 \), \( |r| < 1 \), so the series converges.
Use the formula for the sum of an infinite geometric series \( S = \frac{a}{1 - r} \) to express the sum, where \( a = 3 \) and \( r = -\frac{1}{\pi} \).
Write the sum explicitly as \( S = \frac{3}{1 - \left(-\frac{1}{\pi}\right)} = \frac{3}{1 + \frac{1}{\pi}} \). Simplify this expression to get the sum in a more compact form.

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Key Concepts

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

Geometric Series Definition

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio, r. It has the form ∑ ar^k, where a is the first term and k is the index. Understanding this structure is essential to identify and evaluate the series.
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Convergence Criteria for Infinite Geometric Series

An infinite geometric series converges if and only if the absolute value of the common ratio |r| is less than 1. If |r| ≥ 1, the series diverges. This criterion determines whether the sum approaches a finite value or not.
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Convergence of an Infinite Series

Sum Formula for Convergent Geometric Series

When a geometric series converges, its sum can be calculated using the formula S = a / (1 - r), where a is the first term and r is the common ratio. This formula provides a quick way to find the total sum of infinitely many terms.
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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 = 1 to ∞) (1 + 2 / k)ᵏ

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Textbook Question

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 0 to ∞) (1/4)ᵏ × 5^(3 – k)

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Textbook Question

13–52. Limits of sequences

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


{√(n² + 1) − n} 

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Textbook Question

45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (k = 2 to ∞) (−1)ᵏ · k · (k² + 1) / (k³ − 1)

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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. 

{Use of Tech} aₙ₊₁ = (aₙ⁄₁₁ )+ 50;a₀ = 50

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Textbook Question

13–52. Limits of sequences

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


{2ⁿ⁺¹3⁻ⁿ}

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