15. Power Series

Taylor series and interval of convergence

c. Determine the interval of convergence of the series.

f(x) = 1/x², a=1

Taylor series and interval of convergence

c. Determine the interval of convergence of the series.

f(x) = e⁻ˣ, a=0

Taylor series and interval of convergence

c. Determine the interval of convergence of the series.

f(x)=2/(1−x)³, a=0

Taylor series and interval of convergence

c. Determine the interval of convergence of the series.

f(x) = e²ˣ, a = 0

Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x)=3ˣ, a=0

How are the Taylor polynomials for a function f centered at a related to the Taylor series of the function f centered at a?

Suppose you know the Maclaurin series for f and that it converges to f(x) for |x|<1. How do you find the Maclaurin series for f(x²) and where does it converge?

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