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 11.4.3

Chapter 11, Problem 11.4.3

How would you approximate e⁻⁰ᐧ⁶ using the Taylor series for eˣ?

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Recall the Taylor series expansion for the exponential function \(e^{x}\) centered at 0, which is given by: \[e^{x} = \sum_{n=0}^{\infty} \frac{x^{n}}{n!} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots\]

To approximate \(e^{-0.6}\), substitute \(x = -0.6\) into the series: \[e^{-0.6} = 1 + (-0.6) + \frac{(-0.6)^{2}}{2!} + \frac{(-0.6)^{3}}{3!} + \cdots\]

Decide how many terms of the series you want to use for the approximation. More terms generally mean a more accurate approximation. For example, you might use the first 4 or 5 terms.

Calculate each term up to the chosen number of terms by raising \(-0.6\) to the appropriate power and dividing by the factorial of the term's index. Remember that factorial \(n!\) is the product of all positive integers up to \(n\).

Sum all the calculated terms together to get the approximate value of \(e^{-0.6}\). This sum will be your Taylor series approximation.

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Key Concepts

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

Taylor Series Expansion

The Taylor series expresses a function as an infinite sum of terms calculated from the function's derivatives at a single point, usually zero. For eˣ, the series is ∑(xⁿ/n!) from n=0 to ∞, allowing approximation of eˣ by summing a finite number of terms.

Substitution of Negative Exponents

To approximate e⁻⁰·⁶, substitute x = -0.6 into the Taylor series for eˣ. This involves raising -0.6 to successive powers and dividing by factorial terms, which accounts for the alternating signs and magnitude changes in the series.

Error and Convergence of the Series

The accuracy of the approximation depends on the number of terms used. Taylor series for eˣ converges for all real x, and truncating after several terms yields a close estimate. Understanding the remainder term helps gauge the approximation's precision.

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Suppose the power series ∑ₖ₌₀∞ cₖ(x−a)ᵏ has an interval of convergence of (−3,7]. Find the center a and the radius of convergence R.

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Textbook Question

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Give the interval of convergence for the new series (Theorem 11.4 is useful). Use the Maclaurin series

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√(9 − 9x)

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Textbook Question

The first three Taylor polynomials for f(x)=√(1+x) centered at 0 are p₀ = 1, p₁ = 1+x/2, and p₂ = 1 + x/2 − x²/8. Find three approximations to √1.1.

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{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.

sin x ≈ x − x³/6 on [π/4, π/4]

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Textbook Question

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.

{(eˣ−1)/x if x ≠ 1, 1 if x = 1

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