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 ∞) k¹⁰⁰ / (k + 1)!
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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 10.1.13
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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8:22
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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5:17Arithmetic Sequences - General Formula
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
16–17. {Use of Tech} Periodic savings
Suppose you deposit m dollars at the beginning of every month in a savings account that earns a monthly interest rate of r, which is the annual interest rate divided by 12 (for example, if the annual interest rate is 2.4%, r = 0.024/12 = 0.002). For an initial investment of m dollars, the amount of money in your account at the beginning of the second month is the sum of your second deposit and your initial deposit plus interest, or m + m(1 + r). Continuing in this fashion, it can be shown that the amount of money in your account after n months is
Aₙ = m + m(1 + r) + ⋯ + m(1 + r)ⁿ⁻¹.
Use geometric sums to determine the amount of money in your savings account after 5 years (60 months) using the given monthly deposit and interest rate.
17. Monthly deposits of \$250 at a monthly interest rate of 0.2%
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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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Textbook Question
Why does the value of a converging alternating series with terms that are nonincreasing in magnitude lie between any two consecutive terms of its sequence of partial sums?
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Textbook Question
39–44. {Use of Tech} Estimating infinite series Estimate the value of the following convergent series with an absolute error less than 10⁻³.
∑ (k = 1 to ∞) (−1)ᵏ / k⁵
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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))
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