Domanda del libro di testo
{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
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Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 11, Problema 11.4.78b
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.
Guida verificata passo dopo passo1
Recall that the sine integral function is defined as \(\mathrm{Si}(x) = \int_0^x f(t) \, dt\), where \(f(t) = \frac{\sin t}{t}\) for \(t \neq 0\) and \(f(0) = 1\).
Start by expressing \(\sin t\) as its Taylor series expansion around \(t=0\):
\(\sin t = \sum_{n=0}^\infty (-1)^n \frac{t^{2n+1}}{(2n+1)!}\).
Divide the series for \(\sin t\) by \(t\) to get the series for \(f(t) = \frac{\sin t}{t}\):
\(f(t) = \sum_{n=0}^\infty (-1)^n \frac{t^{2n}}{(2n+1)!}\).
Integrate the series term-by-term from 0 to \(x\) to find the Taylor series for \(\mathrm{Si}(x)\):
\(\mathrm{Si}(x) = \int_0^x f(t) \, dt = \int_0^x \sum_{n=0}^\infty (-1)^n \frac{t^{2n}}{(2n+1)!} \, dt\).
Interchange the integral and the summation (justified by uniform convergence on compact intervals) and integrate each term:
\(\mathrm{Si}(x) = \sum_{n=0}^\infty (-1)^n \frac{1}{(2n+1)!} \int_0^x t^{2n} \, dt = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)(2n+1)!}\).
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Definition and Properties of the Sine Integral Function
The sine integral function Si(x) is defined as the integral from 0 to x of (sin t)/t dt, with a special value at t=0 to ensure continuity. Understanding this function involves recognizing it as an integral of a function with a removable discontinuity and its role in analysis and applications.
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Percorso guidato
05:43
Definition of the Definite Integral
Taylor Series Expansion of Functions
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point, usually zero. To find the Taylor series of Si(x), one must express the integrand as a power series and then integrate term-by-term, ensuring convergence within the radius of expansion.
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Term-by-Term Integration of Power Series
Integrating a power series term-by-term is valid within the interval of convergence and allows finding the integral of complex functions by integrating each term individually. This technique simplifies finding the Taylor series of integral-defined functions like Si(x) by integrating the series expansion of (sin t)/t.
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05:58Intro to Power Series
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{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
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Domanda del libro di testo
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