Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.6.35

Chapter 10, Problem 10.6.35

33–38. {Use of Tech} Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude. Although you do not need it, the exact value of the series is given in each case.

π / 4 = ∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

Verified step by step guidance

Recognize that the given series is an alternating series of the form \(\sum_{k=0}^\infty (-1)^k \frac{1}{2k+1}\), which converges to \(\frac{\pi}{4}\).

Recall the Alternating Series Remainder Theorem, which states that the magnitude of the remainder \(R_n\) after summing \(n\) terms is less than or equal to the absolute value of the first omitted term: \(|R_n| \leq \left| a_{n+1} \right|\).

Identify the \((n+1)\)-th term of the series: \(a_{n+1} = \frac{1}{2(n+1) + 1} = \frac{1}{2n + 3}\).

Set up the inequality to ensure the remainder is less than \(10^{-4}\): \(\frac{1}{2n + 3} < 10^{-4}\).

Solve this inequality for \(n\) to find the minimum number of terms needed: multiply both sides by \(2n + 3\), then isolate \(n\) and solve for the smallest integer \(n\) satisfying the inequality.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Alternating Series and Convergence

An alternating series is a series whose terms alternate in sign, typically of the form (−1)^k * a_k with a_k > 0. Such series can converge if the terms decrease in magnitude to zero. Understanding this helps determine when the infinite sum approaches a finite limit.

Alternating Series Remainder (Error) Estimation

The remainder after summing n terms of a convergent alternating series is less than or equal to the magnitude of the first omitted term. This property allows us to estimate how many terms are needed to ensure the error is below a desired threshold.

Partial Sums and Error Bounds

A partial sum is the sum of the first n terms of a series. For alternating series, the error bound is given by the absolute value of the (n+1)-th term. Using this, one can find the minimum n such that the remainder is less than a specified small number, like 10⁻⁴.

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Textbook Question

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 1 to ∞) ((1/3) × (5/6)ᵏ + (3/5) × (7/9)ᵏ)

Textbook Question

55–70. More sequences

Find the limit of the following sequences or determine that the sequence diverges.

{(75n⁻¹ / 99ⁿ) + (5ⁿsinn / 8ⁿ)}

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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.

∑ (k = 1 to ∞) 1 / (2k − √k)

Textbook Question

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.

∑ (k = 0 to ∞) k / (2k + 1)

Textbook Question

9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.

∑ (k = 2 to ∞) k / ln k

Textbook Question

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

∑ (from k = 1 to ∞)(ln²k) / k³ᐟ²