15. Power Series

Limits with a parameter Use Taylor series to evaluate the following limits. Express the result in terms of the nonzero real parameter(s).

lim ₓ→₀ (eᵃˣ − 1)/x

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

{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

Find a Taylor series for f centered at 2 given that f⁽ᵏ⁾(2)=1, for all nonnegative integers k.

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)

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.

Find the Taylor polynomials p₁, …, p₅ centered at a=0 for f(x)=e⁻ˣ

Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x) = (1 + x²)⁻¹, a = 0

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

e. The Taylor series for an even function centered at 0 has only even powers of x.

Use the Taylor series for cos x centered at 0 to verify that lim ₓ→ₐ (1− cos x)/x = 0.

{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.

{Use of Tech} Approximations with Taylor polynomials

a. Approximate the given quantities using Taylor polynomials with n = 3.

b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.

e⁰ᐧ¹²

Approximate $\sin 0.3$ to four decimal places using the third-degree Maclaurin polynomial for $f\left(x\right)=\sin x$.

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.

Limits Evaluate the following limits using Taylor series.

lim ₓ→₀ (sin 2x)/x

Derivative trick Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Then f⁽ᵏ⁾(a)=k! multiplied by the coefficient of (x−a)ᵏ. Use this idea to evaluate f⁽³⁾(0) and f⁽⁴⁾(0) for the following functions. Use known series and do not evaluate derivatives.

f(x) = eᶜᵒˢ ˣ