Textbook Question
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) = e⁻ˣ, a = 0
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Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 11, Problem 11.4.62
Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₁∞ (x²ᵏ)/k
Verified step by step guidance1
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.
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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
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Textbook Question
Limits Evaluate the following limits using Taylor series.
lim ₓ→₀ (eˣ − e⁻ˣ)/x
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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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