Textbook Question
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)5¹⁻²ᵏ
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 10.1.13
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = 1/10ⁿ
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Identify the general term of the sequence given by the explicit formula \(a_n = \frac{1}{10^n}\), where \(n\) is the term number starting from 1.
To find the first term \(a_1\), substitute \(n=1\) into the formula: \(a_1 = \frac{1}{10^1}\).
To find the second term \(a_2\), substitute \(n=2\) into the formula: \(a_2 = \frac{1}{10^2}\).
To find the third term \(a_3\), substitute \(n=3\) into the formula: \(a_3 = \frac{1}{10^3}\).
To find the fourth term \(a_4\), substitute \(n=4\) into the formula: \(a_4 = \frac{1}{10^4}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sequences and Terms
A sequence is an ordered list of numbers defined by a specific rule. Each number in the sequence is called a term, denoted as aₙ, where n indicates the term's position. Understanding how to identify and write terms from a given formula is fundamental.
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Introduction to Sequences
Explicit Formula for a Sequence
An explicit formula defines the nth term of a sequence directly in terms of n, allowing calculation of any term without knowing previous terms. For example, aₙ = 1/10ⁿ means each term is the reciprocal of 10 raised to the nth power.
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Exponentiation and Powers of 10
Exponentiation involves raising a base number to a power, indicating repeated multiplication. Powers of 10 are especially important in sequences, as 10ⁿ means 10 multiplied by itself n times, affecting the size and pattern of the terms.
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05:58Intro to Power Series
Related Practice
Textbook Question
17–22. Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied.
∑ (k = 1 to ∞) 1 / (∛(5k + 3))
Textbook Question
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{bₙ}, where
bₙ = { n / (n + 1)if n ≤ 5000
ne⁻ⁿif n > 5000 }
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Textbook Question
11–27. Alternating Series Test Determine whether the following series converge.
∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k¹/ᵏ
Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
29.∑ (k = 1 to ∞) e^(–2k)
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Textbook Question
21–42. Geometric series Evaluate each geometric series or state that it diverges.
33.∑ (k = 4 to ∞) 1 / 5ᵏ
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