Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

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

Briggs 3rd Edition

Ch. 11 - Power Series

Problem 11.4.76a

Chapter 11, Problem 11.4.76a

Probability: sudden−death playoff Teams A and B go into suddendeath overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Assume each team has a 1/6 chance of scoring when it has the ball, and Team A has the ball first.

a. The probability that Team A ultimately wins is ∑ₖ₌₀∞ (1/6)(5/6)²ᵏ. Evaluate this series.

Verified step by step guidance

Recognize that the given series is an infinite geometric series of the form \(\sum_{k=0}^\infty ar^k\), where the first term \(a = \frac{1}{6}\) and the common ratio \(r = \left(\frac{5}{6}\right)^2\).

Recall the formula for the sum of an infinite geometric series when \(|r| < 1\): \(S = \frac{a}{1 - r}\).

Substitute the values of \(a\) and \(r\) into the formula: \(S = \frac{\frac{1}{6}}{1 - \left(\frac{5}{6}\right)^2}\).

Simplify the denominator by calculating \(\left(\frac{5}{6}\right)^2 = \frac{25}{36}\), so the denominator becomes \(1 - \frac{25}{36}\).

Express the denominator as a single fraction and then write the sum \(S\) as a simplified fraction or decimal (without calculating the final numeric value).

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Key Concepts

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

Geometric Series

A geometric series is a sum of terms where each term is a constant multiple (common ratio) of the previous one. It has the form ∑ ar^k, and if |r| < 1, the series converges to a/(1-r). This concept is essential for evaluating infinite sums like the one given in the problem.

Probability of Independent Events

When events are independent, the probability of their joint occurrence is the product of their individual probabilities. Here, each scoring attempt is independent, so the probability of sequences of successes and failures can be calculated by multiplying the respective probabilities.

Infinite Sum in Probability Context

In problems involving repeated trials until success, probabilities often form infinite sums representing all possible outcomes. Understanding how to express and evaluate these sums helps determine the total probability of an event occurring eventually, such as Team A winning in sudden-death overtime.

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Related Practice

Textbook Question

Taylor series and interval of convergence

a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.

f(x) = tan ⁻¹ (x/2), a = 0

Textbook Question

Matching functions with polynomials Match functions a–f with Taylor polynomials A–F (all centered at 0). Give reasons for your choices.

a. √(1 + 2x)

A. p₂(x)= 1 + 2x + 2x²

B. p₂(x) = 1 − 6x + 24x²

C. p₂(x) = 1 + x − x²/2

D. p₂(x) = 1 − 2x + 4x²

E. p₂(x) = 1 − x + (3/2)x²

F. p₂(x) = 1 − 2x + 2x²

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Textbook Question

Taylor series

a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.

f(x) = ln x, a = 3

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Textbook Question

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

a. The function f(x) = √x has a Taylor series centered at 0.

Textbook Question

Taylor series and interval of convergence

a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.

f(x)=2/(1−x)³, a=0

Textbook Question

Taylor series

a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.

f(x) = 2ˣ, a = 1