Textbook Question
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = e⁻ˣ, a=0
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 11.3.23b
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = cosh 3x, a = 0
Verified step by step guidance1
Recall the definition of the hyperbolic cosine function: \(\cosh x = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}\). This is the Maclaurin series expansion (Taylor series at \(a=0\)) for \(\cosh x\).
Since the function is \(f(x) = \cosh 3x\), substitute \$3x\( in place of \)x$ in the series expansion. This gives \(f(x) = \sum_{n=0}^{\infty} \frac{(3x)^{2n}}{(2n)!}\).
Simplify the power inside the summation: \((3x)^{2n} = 3^{2n} x^{2n}\). So the series becomes \(f(x) = \sum_{n=0}^{\infty} \frac{3^{2n} x^{2n}}{(2n)!}\).
Express the power series explicitly in summation notation centered at \(a=0\) as \(f(x) = \sum_{n=0}^{\infty} \frac{3^{2n}}{(2n)!} (x - 0)^{2n}\).
This is the power series representation of \(\cosh 3x\) about \(a=0\) in summation notation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor Series
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. For a function f(x) centered at a = 0, it is expressed as the sum of (f^n(0)/n!) * x^n, where f^n(0) is the nth derivative evaluated at zero. This series approximates the function near the center point.
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Taylor Series
Power Series and Summation Notation
A power series is an infinite series of the form Σ c_n (x - a)^n, where c_n are coefficients and a is the center. Summation notation compactly expresses this infinite sum using the sigma symbol (Σ), indicating the sum over all terms indexed by n. Writing a function as a power series helps analyze and approximate it.
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Interval of Convergence
The interval of convergence is the set of x-values for which a power series converges to a finite value. It is found by applying convergence tests like the ratio test. Understanding this interval is crucial because the Taylor series only accurately represents the function within this range.
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Related Practice
Textbook Question
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
b. Write the power series using summation notation.
f(x) = 1/x², a=1
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. The function f(x) = csc x has a Taylor series centered at π/2.
Textbook Question
Sine integral function The function Si(x) = ∫₀ˣ f(t) dt, where f(t) = {(sin t)/t if t ≠ 0, 1 if t = 0, is called the sine integral function.
b. Integrate the series to find a Taylor series for Si.
Textbook Question
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
b. Estimate f(0.2) and give a bound on the error in the approximation.
f(x) =√(1+x) ≈ 1 + x/2
Textbook Question
Taylor series
b. Write the power series using summation notation.
f(x)=sin x, a = π/2
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