Textbook Question
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = 1/10ⁿ
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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 10.6.39
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
Verified step by step guidance1
Recognize that the series given is an alternating series of the form \(\sum_{k=1}^\infty \frac{(-1)^k}{k^5}\), where the terms decrease in absolute value and approach zero as \(k\) increases, which means the series converges by the Alternating Series Test.
To estimate the sum with an absolute error less than \(10^{-3}\), use the Alternating Series Estimation Theorem, which states that the absolute error when approximating the sum by the first \(n\) terms is less than or equal to the absolute value of the first omitted term, i.e., \(|R_n| \leq |a_{n+1}|\).
Find the smallest integer \(n\) such that the absolute value of the \((n+1)\)-th term satisfies \(\left| \frac{(-1)^{n+1}}{(n+1)^5} \right| = \frac{1}{(n+1)^5} < 10^{-3}\).
Once \(n\) is found, compute the partial sum \(S_n = \sum_{k=1}^n \frac{(-1)^k}{k^5}\) by adding the first \(n\) terms of the series.
This partial sum \(S_n\) will be an estimate of the infinite series with an absolute error less than \(10^{-3}\), as guaranteed by the Alternating Series Estimation Theorem.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Convergence of Infinite Series
An infinite series converges if the sum of its terms approaches a finite limit as the number of terms increases. For alternating series like this one, the Alternating Series Test helps determine convergence by checking if terms decrease in absolute value and approach zero.
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Convergence of an Infinite Series
Alternating Series Estimation Theorem
This theorem states that the absolute error when approximating an alternating series by its partial sum is less than or equal to the absolute value of the first omitted term. It allows us to estimate how many terms are needed to achieve a desired error bound.
Recommended video:
06:00Geometric Series
Absolute Error and Partial Sums
Absolute error measures the difference between the true sum and the partial sum approximation. By calculating partial sums and comparing the size of the next term, we can ensure the error is below a specified threshold, such as 10⁻³ in this problem.
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04:57Determining Error and Relative Error
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
54–69. Telescoping series
For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.
61. ∑ (k = 1 to ∞) ln((k + 1) / k)
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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
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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