Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

문제 10.6.39

10장, 문제 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

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.

관련 실천

교과서 질문

13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.

aₙ = 1/10ⁿ

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교과서 질문

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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교과서 질문

21–42. Geometric series Evaluate each geometric series or state that it diverges.

33.∑ (k = 4 to ∞) 1 / 5ᵏ

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교과서 질문

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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교과서 질문

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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교과서 질문

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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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

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Convergence of Infinite Series

Alternating Series Estimation Theorem

Absolute Error and Partial Sums

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