15. Power Series

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

The first three Taylor polynomials for f(x)=√(1+x) centered at 0 are p₀ = 1, p₁ = 1+x/2, and p₂ = 1 + x/2 − x²/8. Find three approximations to √1.1.

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).

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.

b. Write the power series using summation notation.

f(x) = 1/x², a=1

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. If f has a Taylor series that converges only on (−2,2), then f(x²) has a Taylor series that also converges only on (−2,2).

Any method

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients.

b. Determine the radius of convergence of the series.

f(x) = (1 + x²)⁻²/³

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

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