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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 10.3.91
Chapter 10, Problem 10.3.91

{Use of Tech} For what value of r does
∑ (k = 3 to ∞) r²ᵏ = 10?

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1
Recognize that the given series is an infinite geometric series starting from k = 3, with the general term \( r^{2k} \). This means the terms are \( r^{6}, r^{8}, r^{10}, \ldots \).
Identify the first term \( a \) of the series as \( r^{6} \) (when \( k=3 \)) and the common ratio \( q \) as \( r^{2} \) because each term increases the exponent by 2.
Recall the formula for the sum of an infinite geometric series \( S = \frac{a}{1 - q} \), which converges if \( |q| < 1 \). Here, the sum is given as 10, so set up the equation \( \frac{r^{6}}{1 - r^{2}} = 10 \).
Multiply both sides of the equation by \( 1 - r^{2} \) to clear the denominator, resulting in \( r^{6} = 10(1 - r^{2}) \).
Rewrite the equation to isolate terms and form a polynomial in terms of \( r \), which can then be solved to find the value(s) of \( r \) that satisfy the equation.

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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 a sum of terms where each term is found by multiplying the previous term by a constant ratio r. The series ∑ r^k from k = n to ∞ converges if |r| < 1, and its sum can be calculated using the formula S = a / (1 - r), where a is the first term.
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Geometric Series

Sum of an Infinite Geometric Series Starting at k = n

When a geometric series starts at an index k = n instead of 0, the first term a is r^n. The sum from k = n to ∞ is S = r^n / (1 - r), assuming |r| < 1. This adjustment is crucial for correctly evaluating series that do not start at zero.
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Geometric Series

Solving for the Common Ratio in an Infinite Series

To find the value of r that satisfies a given sum, set the sum formula equal to the target value and solve the resulting equation. This often involves algebraic manipulation and considering the convergence condition |r| < 1 to ensure the series sum is finite.
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Convergence of an Infinite Series
Related Practice
Textbook Question

What test is advisable if a series involves a factorial term?

Textbook Question

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


∑ (from k = 1 to ∞)(1 / √(k + 2) – 1 / √k)

Textbook Question

Evaluate 1000!/998! without a calculator.

Textbook Question

84–87. {Use of Tech} Sequences by recurrence relations

The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.


a.Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.

b.Use analytical methods to find the limit of the sequence.


aₙ₊₁ = 2aₙ(1 − aₙ);a₀ = 0.3

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Textbook Question

What comparison series would you use with the Limit Comparison Test to determine whether ∑ (k = 1 to ∞) (k² + k + 5) / (k³ + 3k + 1) converges?

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 ∞) ((-1)ᵏ⁺¹) × ((10k³ + k) / (9k³ + k + 1))ᵏ