15. Power Series

Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.

ƒ(x) = tan⁻¹(4x), a = 0

Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.

ƒ(x) = 1/(4 + x²), a = 0

How would you approximate e⁻⁰ᐧ⁶ using the Taylor series for eˣ?

Use the Taylor series for cos x centered at 0 to verify that lim ₓ→ₐ (1− cos x)/x = 0.

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

a. The function f(x) = √x has a Taylor series centered at 0.

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

b. The function f(x) = csc x has a Taylor series centered at π/2.

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

d. If p(x) is the Taylor series for f centered at 0, then p(x−1) is the Taylor series for f centered at 1.

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.

∑ₖ₌₁∞ (3x + 2)ᵏ/k

Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.∫₀⁰ᐧ²⁵ e⁻ˣ² dx

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.

√e

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.

tan ⁻¹ (1/2)

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.

cos 2

{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.

∫₀⁰ᐧ³⁵ tan ⁻¹x dx

{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.

∫₀⁰ᐧ² (ln (1 + t))/t dt

{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.

∫₀⁰ᐧ² sin x² dx