15. Power Series

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)=3ˣ, a=0

{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.

Limits Evaluate the following limits using Taylor series.

lim ₓ→∞ x(e¹/ˣ − 1)

Find the interval of convergence for the Taylor series for $f\left(x\right)=\sin x$ centered at $x=\frac{\pi }{2}$.

{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.

Finding Taylor and Maclaurin Series

In Exercises 25–34, find the Taylor series generated by f at x = a.

f(x) = cos(2x + π/2),a = π/4

Approximating ln 2 Consider the following three ways to approximate

ln 2.

d. At what value of x should the series in part (c) be evaluated to approximate ln 2? Write the resulting infinite series for ln 2.

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.

(1 + x⁴)⁻¹

Approximating ln 2 Consider the following three ways to approximate

ln 2.

c. Use the property ln a/b = ln a - ln b and the series of parts (a) and (b) to find the Taylor series for ƒ(x) = ln (1 + x)/(1 - x) b centered at 0.

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

d. Suppose f'' is continuous on an interval that contains a, where f has an inflection point at a. Then the second−order Taylor polynomial for f at a is linear.

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) = tan ⁻¹ (x/2), a = 0

Matching functions with polynomials Match functions a–f with Taylor polynomials A–F (all centered at 0). Give reasons for your choices.

a. √(1 + 2x)

A. p₂(x)= 1 + 2x + 2x²

B. p₂(x) = 1 − 6x + 24x²

C. p₂(x) = 1 + x − x²/2

D. p₂(x) = 1 − 2x + 4x²

E. p₂(x) = 1 − x + (3/2)x²

F. p₂(x) = 1 − 2x + 2x²

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

Limits Evaluate the following limits using Taylor series.

lim ₓ→₀ (tan ⁻¹ x − x)/x³"

Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x) = cosh 3x, a = 0