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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 11.4.62
Chapter 11, Problem 11.4.62

Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₁∞ (x²ᵏ)/k

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Recognize the given power series: \( \sum_{k=1}^{\infty} \frac{x^{2k}}{k} \). Notice that the powers of \( x \) are even, i.e., \( x^{2k} = (x^2)^k \). This suggests rewriting the series in terms of \( y = x^2 \).
Rewrite the series as \( \sum_{k=1}^{\infty} \frac{y^k}{k} \), where \( y = x^2 \). This is a known power series form related to the natural logarithm function.
Recall the power series expansion for the natural logarithm: \( -\ln(1 - y) = \sum_{k=1}^{\infty} \frac{y^k}{k} \) for \( |y| < 1 \). This matches the rewritten series exactly.
Substitute back \( y = x^2 \) into the expression to identify the function represented by the original series: \( \sum_{k=1}^{\infty} \frac{x^{2k}}{k} = -\ln(1 - x^2) \) for \( |x| < 1 \).
Conclude that the given power series represents the function \( -\ln(1 - x^2) \) within its radius of convergence.

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Key Concepts

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

Power Series Representation of Functions

A power series is an infinite sum of terms in the form a_k(x - c)^k, where a_k are coefficients and c is the center. Many functions can be expressed as power series within a radius of convergence, allowing complex functions to be analyzed and approximated using polynomials.
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Recognizing Known Power Series Forms

Certain standard functions have well-known power series expansions, such as the geometric series or the series for logarithmic and exponential functions. Identifying these forms helps in matching a given series to a known function by comparing coefficients and powers.
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Manipulating Series to Match Standard Forms

To identify a function from a given power series, it is often necessary to rewrite or manipulate the series, such as changing indices, factoring terms, or substituting variables. This process helps transform the series into a recognizable standard form linked to a known function.
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Related Practice
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

How are the Taylor polynomials for a function f centered at a related to the Taylor series of the function f centered at a?

Textbook Question

{Use of Tech} Remainders Let 


f(x) = ∑ₖ₌₀∞ xᵏ = 1/(1−x) and Sₙ(x) = ∑ₖ₌₀ⁿ⁻¹ xᵏ


The remainder in truncating the power series after n terms is Rₙ = f(x) − Sₙ(x), which depends on x.


a. Show that Rₙ(x) = xⁿ /(1−x).

b. Graph the remainder function on the interval |x| < 1, for n=1, 2, and 3 . Discuss and interpret the graph. Where on the interval is |Rₙ(x)| largest? Smallest?

c. For fixed n, minimize |Rₙ(x)| with respect to x. Does the result agree with the observations in part (b)?

d. Let N(x) be the number of terms required to reduce |Rₙ(x)| to less than 10⁻⁶. Graph the function N(x) on the interval |x|<1.

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Textbook Question

{Use of Tech} Graphing Taylor polynomials


a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n=1 and n=2.


b. Graph the Taylor polynomials and the function.


f(x)=sin x, a=π/4

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

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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