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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.R.65e
Chapter 11, Problem 11.R.65e

Approximating ln 2 Consider the following three ways to approximate
ln 2.
e. Using four terms of the series, which of the three series derived in parts (a)–(d) gives the best approximation to ln 2? Can you explain why?

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Recall the three series expansions for \( \ln 2 \) derived in parts (a)–(d). Typically, these might include: the Taylor series expansion of \( \ln(1+x) \) at \( x=1 \), the alternating series expansion, or other related series. Identify each series explicitly before proceeding.
Write down the first four terms of each series. For example, if the series is \( \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \), substitute \( x=1 \) and list the first four terms for each series.
Calculate the partial sums of the first four terms for each series (without simplifying to a final decimal). This means summing the terms as they are, keeping the fractions or expressions intact.
Compare the magnitude of the remainder (error) term for each series after four terms. Use the Alternating Series Estimation Theorem or remainder bounds for Taylor series to estimate which series has the smallest error after four terms.
Explain why the series with the smallest remainder or error term provides the best approximation. Typically, this is because the terms decrease in magnitude faster or the series converges more quickly at \( x=1 \).

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Key Concepts

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

Taylor and Maclaurin Series

Taylor and Maclaurin series express functions as infinite sums of polynomial terms derived from their derivatives at a point. Approximating ln(2) often involves using a Maclaurin series expansion of ln(1+x) with x=1. Understanding how these series are constructed helps in selecting and evaluating approximations.
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Convergence of Taylor & Maclaurin Series

Convergence and Error in Series Approximations

The accuracy of approximating functions using series depends on how quickly the series converges and the size of the remainder (error) after a finite number of terms. Some series converge faster near certain points, making their partial sums better approximations. Comparing errors helps determine which series best approximates ln(2) with four terms.
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Convergence of an Infinite Series

Alternating Series and Error Bounds

Many series for ln(1+x) are alternating series, where terms alternate in sign. The Alternating Series Estimation Theorem states that the error after n terms is less than or equal to the absolute value of the next term. Recognizing this helps explain why some series provide better approximations for ln(2) when truncated.
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Geometric Series
Related Practice
Textbook Question

Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.

ƒ(x) = e^(sin x), n = 2, a = 0

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Textbook Question

Approximating ln 2 Consider the following three ways to approximate

ln 2.

b. Use the Taylor series for ln (1 - x) centered at 0 and the identity ln 2 = -ln 1/2. Write the resulting infinite series.

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Textbook Question

Limits by power series Use Taylor series to evaluate the following limits.


lim ₙ → ₄ ln (x - 3)/(x² - 16)

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Textbook Question

Find the remainder term Rₙ(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.)


ƒ(x) = eˣ; bound R₃(x), for |x| < 1

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Textbook Question

Power series from the geometric series Use the geometric series a Σₖ ₌ ₀ ∞ (x)ᵏ = 1/(1 - x), for |x| < 1, to determine the Maclaurin series and the interval of convergence for the following functions.


ƒ(x) = ln (1 - 4x)

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Textbook Question

ƒ(x) = eˣ, a = 0; e-0.08


b. Use the Taylor polynomials to approximate the given expression. Make a table showing the approximations and the absolute error in these approximations using a calculator for the exact function value.

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