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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 11.1.46
Chapter 11, Problem 11.1.46

Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.


f(x) = 1/(1 - x), a=0

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Identify the function and the center of the Taylor polynomial: here, the function is \(f(x) = \frac{1}{1 - x}\) and the Taylor polynomial is centered at \(a = 0\).
Recall the formula for the remainder (Lagrange form) of the nth-order Taylor polynomial centered at \(a\): \[R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} (x - a)^{n+1}\] where \(c\) is some value between \(a\) and \(x\).
Find the \((n+1)\)th derivative of \(f(x)\). Since \(f(x) = (1 - x)^{-1}\), use the general formula for derivatives of this form: \[f^{(k)}(x) = k! \cdot (1 - x)^{-(k+1)}\] with appropriate sign considerations.
Substitute \(a = 0\) into the remainder formula and express \(R_n(x)\) in terms of \(f^{(n+1)}(c)\) and \((x - 0)^{n+1} = x^{n+1}\).
Write the final expression for the remainder \(R_n(x)\) as: \[R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}\] where \(c\) lies between \(0\) and \(x\), and \(f^{(n+1)}(c)\) is the \((n+1)\)th derivative evaluated at \(c\).

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Key Concepts

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

Taylor Polynomial

A Taylor polynomial of order n approximates a function near a point a using a finite sum of derivatives of the function at a. It is expressed as the sum of terms involving the function's derivatives up to the nth order, each multiplied by powers of (x - a) and divided by factorials.
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Taylor Polynomials

Remainder (Error) Term in Taylor Series

The remainder term Rₙ represents the difference between the actual function value and its nth-order Taylor polynomial approximation. It quantifies the error and can be expressed using forms like Lagrange's form, involving the (n+1)th derivative evaluated at some point between a and x.
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Taylor Series

Derivatives of f(x) = 1/(1 - x)

For f(x) = 1/(1 - x), the nth derivative is n!/(1 - x)^(n+1). Understanding this pattern is essential to formulating the Taylor polynomial and remainder term, as the derivatives determine the coefficients and the behavior of the remainder.
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Derivative of the Natural Exponential Function (e^x)
Related Practice
Textbook Question

{Use of Tech} Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.

f(x) =∛x with a=64; approximate ∛60.

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.


sinh x²

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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) − y = 0, y(0) = 2

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Textbook Question

Limits Evaluate the following limits using Taylor series.

lim ₓ→∞ x sin(1/x)

Textbook Question

In terms of the remainder, what does it mean for a Taylor series for a function f to converge to f?

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.


1/(1 − 2x)

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