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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.4.43a
Capitolo 10, Problema 10.4.43a

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.


a. Find an upper bound for the remainder in terms of n.


43. ∑ (k = 1 to ∞) 1 / 3ᵏ

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1
Recognize that the series \( \sum_{k=1}^{\infty} \frac{1}{3^k} \) is a geometric series with the first term \( a = \frac{1}{3} \) and common ratio \( r = \frac{1}{3} \).
Recall that for a convergent geometric series with \( |r| < 1 \), the sum to infinity is \( S = \frac{a}{1 - r} \).
The remainder \( R_n \) after summing the first \( n \) terms is the difference between the total sum and the partial sum: \( R_n = S - S_n \).
The partial sum of the first \( n \) terms of a geometric series is \( S_n = a \frac{1 - r^n}{1 - r} \). Substitute \( a = \frac{1}{3} \) and \( r = \frac{1}{3} \) to express \( S_n \).
Express the remainder \( R_n \) in terms of \( n \) by using the formula \( R_n = S - S_n = \frac{a}{1 - r} - a \frac{1 - r^n}{1 - r} = a \frac{r^n}{1 - r} \). Substitute the values of \( a \) and \( r \) to get the upper bound for the remainder.

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Convergent Geometric Series

A geometric series is a sum of terms where each term is a constant multiple (common ratio) of the previous one. If the absolute value of the common ratio is less than 1, the series converges to a finite sum. For example, the series ∑ (1/3)^k converges because |1/3| < 1.
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Geometric Series

Remainder (Error) in Infinite Series

The remainder after n terms of a convergent series is the difference between the infinite sum and the partial sum up to n terms. It measures the error when approximating the infinite sum by a finite sum. Finding an upper bound for the remainder helps estimate how close the partial sum is to the total.
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Convergence of an Infinite Series

Formula for Remainder in Geometric Series

For a geometric series with first term a and common ratio r (|r|<1), the remainder after n terms is given by R_n = a * r^(n+1) / (1 - r). This formula provides an explicit upper bound for the error when approximating the infinite sum by the first n terms.
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Geometric Series
Pratica correlata
Domanda del libro di testo

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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Domanda del libro di testo

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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Domanda del libro di testo

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


a.Write out the first five terms of the sequence.


Drug elimination

Jack took a 200-mg dose of a pain killer at midnight. Every hour, 5% of the drug is washed out of his bloodstream. Let dₙ be the amount of drug in Jack’s blood n hours after the drug was taken, where d₀ = 200mg.

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Domanda del libro di testo

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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Domanda del libro di testo

{Use of Tech} Periodic dosing

Many people take aspirin on a regular basis as a preventive measure for heart disease. Suppose a person takes 80 mg of aspirin every 24 hours. Assume aspirin has a half-life of 24 hours; that is, every 24 hours, half of the drug in the blood is eliminated.


a.Find a recurrence relation for the sequence {dₙ} that gives the amount of drug in the blood after the nᵗʰ dose, where d₁ = 80.

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Domanda del libro di testo

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


a. A series that converges must converge absolutely.

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