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 11.4.11
Limits Evaluate the following limits using Taylor series.
lim ₓ→₀ (eˣ − e⁻ˣ)/x
Verified step by step guidance1
Recall the Taylor series expansions of the exponential functions around 0: \(e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots\) and \(e^{-x} = 1 - x + \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + \cdots\).
Substitute these expansions into the expression \(\frac{e^{x} - e^{-x}}{x}\) to get \(\frac{\left(1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots\right) - \left(1 - x + \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + \cdots\right)}{x}\).
Simplify the numerator by combining like terms: the constant terms cancel out, and you combine the \(x\) terms, \(x^{2}\) terms, etc., carefully noting the signs.
After simplification, factor out \(x\) from the numerator to cancel with the denominator, which will help in evaluating the limit as \(x\) approaches 0.
Evaluate the resulting expression by substituting \(x = 0\) into the simplified series to find the limit.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor Series Expansion
A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. It approximates functions near that point, allowing complex expressions to be simplified into polynomials. For example, eˣ can be expanded around x=0 as 1 + x + x²/2! + x³/3! + ....
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Limit Evaluation Using Series
When direct substitution in a limit leads to an indeterminate form, expanding functions into their Taylor series can simplify the expression. By substituting series expansions, terms can often be canceled or simplified, making it easier to find the limit as x approaches a point.
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Hyperbolic Sine Function and Its Relation
The expression (eˣ − e⁻ˣ)/2 defines the hyperbolic sine function, sinh(x). Recognizing this helps simplify the limit problem since (eˣ − e⁻ˣ)/x = 2 * (sinh(x)/x). Understanding sinh(x) and its series expansion aids in evaluating limits involving exponential differences.
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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
Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₁∞ (x²ᵏ)/k
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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
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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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