Textbook Question
Find a Taylor series for f centered at 2 given that f⁽ᵏ⁾(2)=1, for all nonnegative integers k.
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 11.2.39
Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
ƒ(x) = tan⁻¹(4x), a = 0
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Recall that the Taylor series of a function \( f(x) \) centered at \( a \) is given by \( f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x - a)^n \). Here, \( a = 0 \), so the series is centered at zero (Maclaurin series).
Identify the function: \( f(x) = \tan^{-1}(4x) \). We can use the known Maclaurin series for \( \tan^{-1}(x) = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1} \) for \( |x| < 1 \).
Substitute \( 4x \) in place of \( x \) in the series to get \( \tan^{-1}(4x) = \sum_{n=0}^\infty (-1)^n \frac{(4x)^{2n+1}}{2n+1} = \sum_{n=0}^\infty (-1)^n \frac{4^{2n+1} x^{2n+1}}{2n+1} \).
Write out the first three nonzero terms by plugging in \( n = 0, 1, 2 \):
- For \( n=0 \): \( \frac{4^1 x^1}{1} = 4x \)
- For \( n=1 \): \( - \frac{4^3 x^3}{3} = - \frac{64 x^3}{3} \)
- For \( n=2 \): \( \frac{4^5 x^5}{5} = \frac{1024 x^5}{5} \).
Finally, write the Taylor series in summation notation as \( \sum_{n=0}^\infty (-1)^n \frac{4^{2n+1}}{2n+1} x^{2n+1} \).
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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 derivatives of the function at a single point. Each term involves the nth derivative evaluated at the center point, multiplied by (x - a)^n and divided by n!. This allows approximation of functions near the center point.
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Derivatives of Inverse Trigonometric Functions
Understanding how to differentiate inverse trigonometric functions like arctan(x) is essential. For example, the derivative of arctan(x) is 1/(1 + x^2). When the function is arctan(4x), the chain rule applies, multiplying by the derivative of 4x, which is 4.
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Summation Notation for Series
Summation notation concisely expresses infinite series using the sigma symbol (∑). It includes an index of summation, limits, and a general term formula. Writing the Taylor series in summation form captures all terms compactly and highlights the pattern in coefficients and powers.
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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.
f(x) = 8x^(3/2), a=1; approximate 8 ⋅ 1.1^(3/2)
Textbook Question
{Use of Tech} Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.
ln 1.04, n=3
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 sin x² dx
Textbook Question
{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.
√(1+x) ≈ 1 + x/2 on [−0.1,0.1]
Textbook Question
Probability: tossing for a head The expected (average) number of tosses of a fair coin required to obtain the first head is ∑ₖ₌₁∞ k(1/2)ᵏ. Evaluate this series and determine the expected number of tosses. (Hint: Differentiate a geometric series.)