15. Power Series

Find the Taylor polynomials of order $0,1,2$, and $3$ for $f\left(x\right)=\ln \left(x\right)$ centered at $x=1$.

Taylor series and interval of convergence

c. Determine the interval of convergence of the series.

f(x) = 1/x², a=1

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) = sin x, a = 0

Remainders Find the remainder in the Taylor series centered at the point a for the following functions. Then show that lim ₙ→∞ Rₙ(x)=0, for all x in the interval of convergence.

f(x) = e⁻ˣ, a = 0

Approximating ln 2 Consider the following three ways to approximate

ln 2.

e. Using four terms of the series, which of the three series derived in parts (a)–(d) gives the best approximation to ln 2? Can you explain why?

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

ƒ(x) = e^(sin x), n = 2, a = 0

Taylor series and interval of convergence

c. Determine the interval of convergence of the series.

f(x) = e²ˣ, a = 0

Taylor series and interval of convergence

c. Determine the interval of convergence of the series.

f(x)=3ˣ, a=0

Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x) = ln (x − 2), a = 3

Error Estimates

The approximation eˣ = 1 + x + (x² / 2) is used when x is small. Use the Remainder Estimation Theorem to estimate the error when |x| < 0.1.

Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x)=2/(1−x)³, a=0

Suppose you know the Maclaurin series for f and that it converges to f(x) for |x|<1. How do you find the Maclaurin series for f(x²) and where does it converge?

{Use of Tech} A different kind of approximation When approximating a function f using a Taylor polynomial, we use information about f and its derivative at one point. An alternative approach (called interpolation) uses information about f at several different points. Suppose we wish to approximate f(x)=sin x on the interval [0, π].

a. Write the (quadratic) Taylor polynomial p₂ for f centered at π/2.

b. Now consider a quadratic interpolating polynomial q(x) = ax² + bx + c. The coefficients a, b, and c are chosen such that the following conditions are satisfied:

q(0) = f(0), q(π/2) = f(π/2), and q(π) = f(π)

Show that q(x) = −(4/π²)x² + (4/π)x.

c. Graph f, p₂, and q on [0, π].

d. Find the error in approximating f(x) = sin x at the points π/4, π/2, 3π/4, and π using p₂ and q.

e. Which function, p₂ or q, is a better approximation to f on [0, π]? Explain.

Symmetry

b. Use infinite series to show that sin x is an odd function. That is, show sin (-x) = -sin 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.

ln 1.04, n=3