15. Power Series

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) = e⁻ˣ, a=0

{Use of Tech} Best center point Suppose you wish to approximate cos (π/ 2) using Taylor polynomials. Is the approximation more accurate if you use Taylor polynomials centered at 0 or at π/6? Use a calculator for numerical experiments and check for consistency with Theorem 11.2. Does the answer depend on the order of the polynomial?

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) = (1 − x)⁻¹

{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) = eˣ ≈ 1 + x

Suppose f(0)=1, f'(0)=0, f''(0)=2, and f⁽³⁾(0)=6. Find the third-order Taylor polynomial for f centered at 0 and use it to approximate f(0.2).

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.

√e

Does the accuracy of an approximation given by a Taylor polynomial generally increase or decrease with the order of the approximation? Explain.

Definite integrals by power series Use a Taylor series to approximate the following definite integrals. Retain as many terms as necessary to ensure the error is less than 10⁻³.

∫₀1 x cos x dx

Limits Evaluate the following limits using Taylor series.

lim ₓ→₄ (x² 16)/(ln (x 3)}

Binomial series Write out the first three terms of the Maclaurin series for the following functions.

ƒ(x) = (1 + 2x)^(-5)

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.∫₀⁰ᐧ²⁵ e⁻ˣ² dx

{Use of Tech} Graphing Taylor polynomials

a. Find the nth-order Taylor polynomials for the following functions centered at the given point a, for n=1 and n=2.

b. Graph the Taylor polynomials and the function.

f(x)=sin x, a=π/4

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) = cos x, a = π/2

Limits Evaluate the following limits using Taylor series.

lim ₓ→₀ (eˣ − e⁻ˣ)/x

Convergence Write the remainder term Rₙ(x) for the Taylor series for the following functions centered at the given point a. Then show that lim ₙ → ∞ |Rₙ(x)| = 0, for all x in the given interval.

ƒ(x) = sinh x + cosh x, a = 0, - ∞ < x < ∞