15. Power Series

Taylor series and interval of convergence

c. Determine the interval of convergence of the series.

f(x) = ln (x − 2), a = 3

Use power series operations to find the Taylor series at x = 0 for the functions in Exercises 13–30.

sin x – x + (x³ / 3!)

Find the interval of convergence for the Maclaurin series for $f\left(x\right)={\tan }^{-1}x$

{Use of Tech} Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.

e⁰ᐧ²⁵, n=4

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.

f(x) = cos x, a = π/4; approximate cos (0.24π)

Taylor series and interval of convergence

c. Determine the interval of convergence of the series.

f(x) = e⁻ˣ, a=0

Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.

ƒ(x) = cos⁻¹ x, n = 2, a = 1/2

Approximating ln 2 Consider the following three ways to approximate

ln 2.

a. Use the Taylor series for ln (1 + x) centered at 0 and evaluate it at x = 1 (convergence was asserted in Table 11.5). Write the resulting infinite series.