Textbook Question
Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₁∞ (x²ᵏ)/k
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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 11.1.43
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
Verified step by step guidance1
Identify the function and the center of the Taylor polynomial: here, the function is \(f(x) = e^{-x}\) and the Taylor polynomial is centered at \(a = 0\).
Recall the general form of the remainder (Lagrange form) for 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 \(f(x) = e^{-x}\). Note that derivatives of \(e^{-x}\) alternate signs but remain proportional to \(e^{-x}\). Express \(f^{(n+1)}(x)\) in terms of \(e^{-x}\) and a sign factor.
Substitute \(a = 0\) and the expression for \(f^{(n+1)}(c)\) into the remainder formula to get:
\[R_n(x) = \frac{(-1)^{n+1} e^{-c}}{(n+1)!} x^{n+1}\]
where \(c\) is between \(0\) and \(x\).
Interpret this remainder term as the error bound for approximating \(f(x)\) by its nth-order Taylor polynomial at \(a=0\), with the exact value depending on the unknown \(c\) in the interval between \(0\) and \(x\).
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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 approximates a function near a point a by using the function's derivatives at that point. The nth-order Taylor polynomial includes terms up to the nth derivative, providing a polynomial approximation that becomes more accurate as n increases.
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Taylor Polynomials
Remainder Term (Lagrange Form)
The remainder term Rₙ measures the error between the actual function and its nth-order Taylor polynomial. In Lagrange form, it is expressed using the (n+1)th derivative evaluated at some point between a and x, multiplied by (x - a)^(n+1) divided by (n+1)!.
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06:32Alternating Series Remainder
Derivatives of the Exponential Function
For f(x) = e^{-x}, all derivatives follow a predictable pattern: the kth derivative is (-1)^k e^{-x}. This property simplifies finding the remainder term since the (n+1)th derivative can be directly expressed, aiding in the general formula for Rₙ.
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04:50Derivatives of General Exponential Functions
Related Practice
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} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.
∫₀⁰ᐧ² (ln (1 + t))/t dt
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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.
Find the Taylor polynomial p₃ centered at a=e for f(x)=ln x.
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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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