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 ∞) (k² / 4ᵏ)
Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 10, Problem 10.3.11
9–15. Geometric sums Evaluate each geometric sum.
{Use of Tech}∑ k = 0 to 20(2/5)²ᵏ
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Identify the type of series given. The sum is a geometric series because each term is obtained by multiplying the previous term by a constant ratio.
Write down the general form of a geometric series sum: \(S_n = a \frac{1 - r^{n+1}}{1 - r}\), where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms minus one.
Determine the first term \(a\) by substituting \(k=0\) into the term expression: \(a = \left(\frac{2}{5}\right)^{2 \cdot 0} = \left(\frac{2}{5}\right)^0\).
Identify the common ratio \(r\) by examining the factor that each term is multiplied by to get the next term. Since the exponent increases by 2 each time, \(r = \left(\frac{2}{5}\right)^2\).
Substitute \(a\), \(r\), and \(n=20\) into the geometric sum formula and simplify the expression to find the sum 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. The series has the form a + ar + ar² + ... + arⁿ, where a is the first term and r is the common ratio.
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Geometric Series
Sum Formula for 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), provided r ≠ 1. This formula allows quick calculation of the sum without adding each term individually.
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Evaluating Powers and Substitution
To evaluate the geometric sum, you must correctly compute powers of the common ratio and substitute values into the sum formula. Understanding exponentiation and careful substitution ensures accurate calculation of the sum.
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Related Practice
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 ∞) k^(–1/5)
Textbook Question
Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.
aₙ = (6ⁿ + 3ⁿ) / (6ⁿ + n¹⁰⁰)
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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
Textbook Question
72–86. Evaluating series Evaluate each series or state that it diverges.
∑ (k = 0 to ∞) (1/4)ᵏ × 5^(3 – k)
Textbook Question
32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (from k = 1 to ∞) (2k + 1)! / (k!)²