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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 11 - Power SeriesProblema 11.4.78a
Capitolo 11, Problema 11.4.78a

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.
a. Expand the integrand in a Taylor series centered at 0.

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Recall the definition of the sine integral function: \(\mathrm{Si}(x) = \int_0^x f(t) \, dt\), where \(f(t) = \frac{\sin t}{t}\) for \(t \neq 0\) and \(f(0) = 1\).
To find the Taylor series expansion of the integrand \(f(t)\) centered at 0, start with the Taylor series expansion of \(\sin t\) around 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 \(f(t) = \frac{\sin t}{t}\): \(f(t) = \sum_{n=0}^\infty (-1)^n \frac{t^{2n}}{(2n+1)!}\).
Note that this series is valid for all \(t\) including \(t=0\) because the term for \(n=0\) is \(\frac{t^0}{1!} = 1\), which matches the given \(f(0) = 1\).
Thus, the Taylor series expansion of the integrand \(f(t)\) centered at 0 is \(f(t) = 1 - \frac{t^2}{3!} + \frac{t^4}{5!} - \frac{t^6}{7!} + \cdots\).

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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. It is important to understand this function as an example of an integral involving a non-elementary integrand that requires special handling at singular points.
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Graph of Sine and Cosine Function

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point, usually zero. Expanding the integrand (sin t)/t in a Taylor series centered at 0 involves expressing sin t as its power series and dividing by t, carefully handling the limit at t = 0.
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Taylor Series

Handling Indeterminate Forms and Continuity

Since (sin t)/t is undefined at t = 0, understanding limits and continuity is essential. Using the limit lim_{t→0} (sin t)/t = 1 ensures the function is well-defined and continuous at zero, which is crucial for correctly expanding the integrand in a Taylor series.
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Intro to Continuity
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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.


a. The probability that Team A ultimately wins is ∑ₖ₌₀∞ (1/6)(5/6)²ᵏ. Evaluate this series.

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{Use of Tech} Binomial series


a. Find the first four nonzero terms of the binomial series centered at 0 for the given function.


f(x) = (1+x)⁻²/³; approximate 1.18⁻²/³.

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Taylor series


a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.


f(x) = 1/x, a = 1

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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.


f(x) = cosh 3x, a = 0

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{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

a. Compute S′(x) and C′(x).

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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. Only even powers of x appear in the Taylor polynomials for f(x)=e⁻²ˣ centered at 0.

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