15. Power Series

Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.

f(x) = 1/(1 - x), a=0

Finding Taylor Series at x = 0 (Maclaurin Series)

Find the Maclaurin series for the functions in Exercises 11–24.

5 cos πx

Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x) = e⁻ˣ, a=0

{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero. 

a. Estimate f(0.1) and give a bound on the error in the approximation.

f(x) = tan⁻¹ x ≈ x

Limits Evaluate the following limits using Taylor series.

lim ₓ→∞ x sin(1/x)

Find the remainder term Rₙ(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.)

ƒ(x) = ln (1 - x); bound R₃(x), for |x| < 1/2

ƒ(x) = eˣ, a = 0; e-0.08

b. Use the Taylor polynomials to approximate the given expression. Make a table showing the approximations and the absolute error in these approximations using a calculator for the exact function value.

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) = ∫₀ˣ sin t² dt

{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 ⁻¹ x ≈ x

A differential equation Find a power series solution of the differential equation y'(x) - 4y + 12 = 0, subject to the condition y(0) = 4. Identify the solution in terms of known functions.

Power series for derivatives

a. Differentiate the Taylor series centered at 0 for the following functions.

b. Identify the function represented by the differentiated series.

c. Give the interval of convergence of the power series for the derivative.

f(x) = eˣ

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.

c. Approximate Si(0.5) and Si(1). Use enough terms of the series so the error in the approximation does not exceed 10⁻³.

In terms of the remainder, what does it mean for a Taylor series for a function f to converge to f?

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.

1/(1 − 2x)

Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x)=3ˣ, a=0