Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problema 10.1.67a

Capitolo 10, Problema 10.1.67a

67–70. Formulas for sequences of partial sums Consider the following infinite series.

a.Find the first four partial sums S₁, S₂, S₃, S₄ of the series.

∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]

Guida verificata passo dopo passo

Identify the general term of the series: \( a_k = \frac{2}{(2k - 1)(2k + 1)} \).

Use partial fraction decomposition to rewrite \( a_k \) in a form that allows telescoping. Set \( \frac{2}{(2k - 1)(2k + 1)} = \frac{A}{2k - 1} + \frac{B}{2k + 1} \) and solve for constants \( A \) and \( B \).

Express each partial sum \( S_n = \sum_{k=1}^n a_k \) by substituting the decomposed form of \( a_k \) and write out the sum explicitly to observe cancellation of terms.

Calculate the first four partial sums \( S_1, S_2, S_3, S_4 \) by summing the first 1, 2, 3, and 4 terms respectively, using the telescoping form to simplify the sums.

Write each partial sum \( S_n \) in its simplified form after cancellation, which will help in understanding the behavior of the series as \( n \) increases.

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Partial Sums of a Series

Partial sums are the sums of the first n terms of a series, denoted as Sₙ = a₁ + a₂ + ... + aₙ. They help analyze the behavior of infinite series by approximating the total sum and are essential for understanding convergence and series evaluation.

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06:45

Intro to Series: Partial Sums

Telescoping Series

A telescoping series is one where many terms cancel out when partial sums are expanded, simplifying the sum significantly. Recognizing telescoping patterns allows easier computation of partial sums and limits, often by expressing terms as differences of fractions.

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06:00

Geometric Series

Decomposition into Partial Fractions

Partial fraction decomposition breaks a complex rational expression into simpler fractions that are easier to sum or integrate. In series, this technique often reveals telescoping behavior by rewriting terms to highlight cancellations.

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10:07

Partial Fraction Decomposition: Distinct Linear Factors

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18–20. Evaluating geometric series two ways Evaluate each geometric series two ways.

a. Find the nth partial sum Sₙ of the series and evaluate lim (as n → ∞) Sₙ.

∑ (k = 0 to ∞) (–2/7)ᵏ

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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. (2n)! / (2n − 1)! = 2n

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57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.

a. Find the first four terms of the sequence of heights {hₙ}.

h₀ = 30,r = 0.25

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87. Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

b. If ∑ (k = 1 to ∞) aₖ diverges, then ∑ (k = 10 to ∞) aₖ diverges.

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Find the first term a and the ratio r of each geometric series.

a. ∑ k = 0 to ∞(2/3) × (1/5)ᵏ

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