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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Related Practice

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

√(1 + x) = 1 + x/2 − x²/8 + x³/16 − ⋯, −1 ≤ x ≤ 1.

√(9 − 9x)

Textbook Question

Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.

f(x) = ln √(1 − x²)

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Textbook Question

Combining power series Use the power series representation

f(x ) =ln (1 − x) = −∑ₖ₌₁∞ xᵏ/k, for −1 ≤ x < 1,

to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.

f(3x) = ln (1 − 3x)

Textbook Question

Differential equations

a. Find a power series for the solution of the following differential equations, subject to the given initial condition

b. Identify the function represented by the power series.

y′(t) − 3y = 10, y(0) = 2

Textbook Question

Is ∑ₖ₌₀ ∞ (5x − 20)ᵏ a power series? If so, find the center a of the power series and state a formula for the coefficients cₖ of the power series.

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Textbook Question

Use of Tech Linear and quadratic approximation

a. Find the linear approximating polynomial for the following functions centered at the given point a.

b. Find the quadratic approximating polynomial for the following functions centered at a.

c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.

f(x)=e⁻²ˣ, a=0; approximate e⁻⁰ᐧ².