Textbook Question
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x)=2/(1−x)³, 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 11.3.31b
Taylor series
b. Write the power series using summation notation.
f(x) = ln x, a = 3
Verified step by step guidance1
Recall that the Taylor series of a function \( f(x) \) centered at \( a \) is given by the formula:
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n \]
where \( f^{(n)}(a) \) is the \( n \)-th derivative of \( f \) evaluated at \( a \).
Identify the function and the center: here, \( f(x) = \ln x \) and \( a = 3 \). We need to find the derivatives of \( \ln x \) evaluated at \( x = 3 \).
Compute the first few derivatives of \( f(x) = \ln x \):
- \( f(x) = \ln x \)
- \( f'(x) = \frac{1}{x} \)
- \( f''(x) = -\frac{1}{x^2} \)
- \( f^{(3)}(x) = \frac{2}{x^3} \)
- \( f^{(4)}(x) = -\frac{6}{x^4} \)
Notice the pattern in the derivatives and their signs.
Evaluate each derivative at \( x = 3 \) to get \( f^{(n)}(3) \). For example, \( f'(3) = \frac{1}{3} \), \( f''(3) = -\frac{1}{9} \), and so on.
Write the Taylor series in summation notation using the derivatives evaluated at \( a = 3 \):
\[ \ln x = \sum_{n=0}^{\infty} \frac{f^{(n)}(3)}{n!} (x - 3)^n \]
where each \( f^{(n)}(3) \) is substituted with the corresponding derivative value found in the previous step.
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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. For a function f(x) centered at a point a, the series is given by f(x) = Σ (f⁽ⁿ⁾(a)/n!) (x - a)ⁿ, where n! is factorial and f⁽ⁿ⁾(a) is the nth derivative at a.
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Taylor Series
Derivatives of the Natural Logarithm Function
The function f(x) = ln(x) has derivatives that follow a specific pattern: the first derivative is 1/x, the second is -1/x², the third is 2/x³, and so on, alternating in sign and involving factorial terms. Understanding this pattern is essential to find the coefficients of the Taylor series at a given point.
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Summation Notation for Power Series
Summation notation (Σ) concisely expresses infinite sums, such as power series. Writing a Taylor series in summation form involves identifying the general term that includes the nth derivative, factorial denominator, and (x - a)ⁿ factor, allowing a compact and clear representation of the series.
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Related Practice
Textbook Question
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = (1 + x²)⁻¹, a = 0
Textbook Question
{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
b. Expand sin t² and cos t² in a Maclaurin series, and then integrate to find the first four nonzero terms of the Maclaurin series for S and C.
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) = ln (1 + x) ≈ x − x²/2
1
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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) = sin x ≈ x
Textbook Question
Probability: sudden−death playoff Teams A and B go into suddendeath overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Assume each team has a 1/6 chance of scoring when it has the ball, and Team A has the ball first.
b. The expected number of rounds (possessions by either team) required for the overtime to end is (1/6) ∑ₖ₌₁∞ k(5/6)ᵏ⁻¹. Evaluate this series.