15. Power Series

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

{Use of Tech} Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.

f(x) = ∜x with a=16; approximate ∜13.

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} Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.

f(x) =∛x with a=64; approximate ∛60.

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) = ln (1 + x)

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

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) = ln x, a = 3

Taylor series

b. Write the power series using summation notation.

f(x) = ln x, a = 3

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