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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 11.2.51
Chapter 11, Problem 11.2.51

Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.


g(x) = 2/(1 − 2x)² using f(x) = 1/(1 − 2x)

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Recall the given function f(x) = \(\frac{1}{1 - 2x}\), which can be expressed as a geometric series centered at 0. The general form of a geometric series is \(\sum\)_{n=0}^\(\infty\) r^n = \(\frac{1}{1-r}\) for |r| < 1. Here, r = 2x, so the power series for f(x) is \(\sum\)_{n=0}^\(\infty\) (2x)^n = \(\sum\)_{n=0}^\(\infty\) 2^n x^n, valid for |2x| < 1 or |x| < \(\frac{1}{2}\).
Notice that g(x) = \(\frac{2}{(1 - 2x)^2}\) is related to f(x) by differentiation. Specifically, the derivative of f(x) with respect to x is f'(x) = \(\frac{d}{dx}\) \(\left\)( \(\frac{1}{1 - 2x}\) \(\right\)). Use the chain rule to find f'(x).
Calculate f'(x): Since f(x) = (1 - 2x)^{-1}, then f'(x) = -1 \(\cdot\) (1 - 2x)^{-2} \(\cdot\) (-2) = \(\frac{2}{(1 - 2x)^2}\). This shows that g(x) = f'(x).
Differentiate the power series for f(x) term-by-term to find the power series for g(x). The term-by-term differentiation of \(\sum\)_{n=0}^\(\infty\) 2^n x^n is \(\sum\)_{n=1}^\(\infty\) n 2^n x^{n-1}. This series represents g(x).
Determine the interval of convergence for g(x). Since differentiation does not change the radius of convergence, the interval of convergence for g(x) is the same as for f(x), which is |x| < \(\frac{1}{2}\).

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Key Concepts

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

Power Series Representation

A power series expresses a function as an infinite sum of terms involving powers of the variable, typically centered at a point (here, 0). Understanding how to write functions like f(x) = 1/(1 − 2x) as a power series is essential, as it forms the basis for manipulating and finding related series for functions like g(x).
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Intro to Power Series

Term-by-Term Differentiation and Integration of Power Series

Power series can be differentiated or integrated term-by-term within their interval of convergence. This property allows us to find the power series for g(x) by differentiating or integrating the known series for f(x), enabling the construction of new series representations from existing ones.
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Intro to Power Series

Interval of Convergence

The interval of convergence is the set of x-values for which a power series converges. When differentiating or integrating a power series, the interval of convergence may change or remain the same, so determining this interval for the resulting series is crucial to ensure the series accurately represents the function.
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Interval of Convergence
Related Practice
Textbook Question

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.

cos 2

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.


Find the Taylor polynomials p₁, p₂, and p₃ centered at a=1 for f(x)=x³.

Textbook Question

Representing functions by power series Identify the functions represented by the following power series.

∑ₖ₌₁∞ (x²ᵏ)/k

Textbook Question

Limits Evaluate the following limits using Taylor series.

lim ₓ→₀ (eˣ − e⁻ˣ)/x

Textbook Question

Combining power series Use the geometric series


f(x) = 1/(1-x) = ∑ₖ₌₀∞ xᵏ, for |x| < 1,


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


g(x) = x³/(1 − x)

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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.


p(x) = 2x⁶ ln(1 − x)