15. Power Series

Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x) = e⁻ˣ, a=0

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)=2/(1−x)³, a=0

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) = (1 + x²)⁻¹, a = 0

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) = e²ˣ, a = 0

Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x) = e²ˣ, a = 0

Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x) = (1 + x²)⁻¹, a = 0

Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x)=2/(1−x)³, a=0

Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x) = tan ⁻¹ (x/2), a = 0

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?