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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 10.4.63
Chapter 10, Problem 10.4.63

48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.
∑ (k = 1 to ∞) 3ᵏ⁺² / 5ᵏ

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1
Identify the given series: \( \sum_{k=1}^{\infty} \frac{3^{k+2}}{5^k} \). Notice that the terms involve exponential expressions with base 3 and 5.
Rewrite the general term to simplify the expression: \( \frac{3^{k+2}}{5^k} = 3^2 \cdot \frac{3^k}{5^k} = 9 \cdot \left( \frac{3}{5} \right)^k \). This shows the series is a constant multiple of a geometric series.
Recognize that the series is geometric with common ratio \( r = \frac{3}{5} \). Recall that a geometric series \( \sum ar^k \) converges if and only if \( |r| < 1 \).
Since \( \left| \frac{3}{5} \right| < 1 \), the geometric series converges. Therefore, the original series converges as well.
To find the sum (if needed), use the formula for the sum of a geometric series starting at \( k=1 \): \[ S = a \cdot \frac{r}{1-r} \], where \( a = 9 \) and \( r = \frac{3}{5} \).

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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 series where each term is obtained by multiplying the previous term by a constant ratio. It has the form ∑ ar^k, and it converges if the absolute value of the ratio |r| < 1. Understanding this helps identify if the given series fits this pattern and whether it converges.
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Convergence Tests for Series

Convergence tests, such as the geometric series test, ratio test, and root test, help determine if an infinite series converges or diverges. Applying these tests involves analyzing the behavior of terms as k approaches infinity to conclude about the sum's finiteness.
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Manipulating Series Terms

Rewriting series terms into a recognizable form, such as factoring constants or expressing terms with exponents clearly, is essential. This simplification allows easier application of convergence tests and better insight into the series' structure.
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Related Practice
Textbook Question

"21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.

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Textbook Question

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.


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Textbook Question

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Textbook Question

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


∑ (from k = 1 to ∞) (−k)³ / (3k³ + 2)

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

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Textbook Question

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

∑ (k = 1 to ∞) (−1)ᵏ · tan⁻¹(k) / k³