15. Power Series

Approximating ln 2 Consider the following three ways to approximate

ln 2.

b. Use the Taylor series for ln (1 - x) centered at 0 and the identity ln 2 = -ln 1/2. Write the resulting infinite series.

Evaluating an infinite series Write the Maclaurin series for f(x) = ln (1+x) and find the interval of convergence. Evaluate f(−1/2) to find the value of ∑ₖ₌₁∞ 1/(k 2ᵏ)

Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.

f(x) = e⁻ˣ, a = 0

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

c. Only even powers of x appear in the nth−order Taylor polynomial for f(x)=√(1+x²) centered at 0.

{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero. 

b. Estimate f(0.2) and give a bound on the error in the approximation.

f(x) = ln (1 + x) ≈ x − x²/2

Finding Taylor and Maclaurin Series

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

f(x) = x³ − 2x + 4,a = 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} Small argument approximations Consider the following common approximations when x is near zero. 

b. Estimate f(0.2) and give a bound on the error in the approximation.

f(x) = sin x ≈ x

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).

Use of Tech Linear and quadratic approximation

a. Find the linear approximating polynomial for the following functions centered at the given point a.

b. Find the quadratic approximating polynomial for the following functions centered at a.

c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.

Find the Taylor polynomials p₁, p₂, and p₃ centered at a=1 for f(x)=x³.

Finding Taylor Polynomials

In Exercises 1–10, find the Taylor polynomials of orders 0, 1, 2, and 3 generated by f at a.

f(x) = sin x,a = 0

Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x) = e²ˣ, a = 0

Finding Taylor Series

Use substitution (as in Formula (7)) to find the Taylor series at x = 0 of the functions in Exercises 1–12.

e⁻ˣ/²

Limits by power series Use Taylor series to evaluate the following limits.

lim ₙ → ₄ ln (x - 3)/(x² - 16)

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