15. Power Series

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) = 2ˣ, a = 1

{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.

{Use of Tech} Binomial series

b. Use the first four terms of the series to approximate the given quantity.

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

ƒ(x) = eˣ, a = 0; e-0.08

a. Find the Taylor polynomials of order n = 1 and n = 2 for the given functions centered at the given point a.

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.

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) = eˣ; bound R₃(x), for |x| < 1

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

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) = ln (x − 2), a = 3

Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x) = ln (x − 2), a = 3

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Give the interval of convergence for the new series (Theorem 11.4 is useful). Use the Maclaurin series

√(1 + x) = 1 + x/2 − x²/8 + x³/16 − ⋯, −1 ≤ x ≤ 1.

√(9 − 9x)

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series

(1 + x)⁻² = 1 − 2x + 3x² − 4x³ + ⋯, for −1 < x < 1.

(1 + 4x)⁻²

Taylor series and interval of convergence

c. Determine the interval of convergence of the series.

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

Taylor series and interval of convergence

c. Determine the interval of convergence of the series.

f(x) = ln (x − 2), a = 3

Taylor series

b. Write the power series using summation notation.

f(x)=sin x, a = π/2

Taylor series and interval of convergence

b. Write the power series using summation notation.

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