Textbook Question
What test is advisable if a series involves a factorial term?
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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 10.6.49
45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (3/4)ᵏ
Verified step by step guidance1
Identify the given series: \( \sum_{k=1}^{\infty} \left( \frac{3}{4} \right)^k \). This is a geometric series with common ratio \( r = \frac{3}{4} \).
Recall that a geometric series \( \sum_{k=1}^{\infty} ar^{k} \) converges if and only if \( |r| < 1 \). Since \( \left| \frac{3}{4} \right| = 0.75 < 1 \), the series converges.
Because the terms \( \left( \frac{3}{4} \right)^k \) are positive, the series is absolutely convergent. Absolute convergence means the series of absolute values \( \sum |a_k| \) converges, which is true here since all terms are positive.
To confirm, write the sum of the geometric series using the formula for \( |r| < 1 \): \[ S = \frac{a}{1 - r} \], where \( a = \left( \frac{3}{4} \right)^1 = \frac{3}{4} \).
Summarize: Since the series converges absolutely, it also converges conditionally by default, but the stronger conclusion is absolute convergence.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Convergence
A series ∑a_k converges absolutely if the series of absolute values ∑|a_k| converges. Absolute convergence guarantees convergence regardless of the sign of terms, and it implies the original series converges. This concept helps distinguish between strong convergence and conditional cases.
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Geometric Series and Its Convergence
A geometric series has the form ∑r^k, where r is a constant ratio. It converges if and only if |r| < 1, and its sum is given by a/(1-r) for the first term a. Recognizing a series as geometric simplifies determining convergence and sum.
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Conditional Convergence
A series converges conditionally if it converges but does not converge absolutely. This means the series ∑a_k converges, but ∑|a_k| diverges. Conditional convergence often occurs in alternating series and requires careful analysis beyond absolute value tests.
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Related Practice
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