15. Power Series

Approximate $\ln 1.5$ to four decimal places using the third-degree Taylor polynomial for $f\left(x\right)=\ln x$.

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) = cosh 3x, a = 0

Taylor series

b. Write the power series using summation notation.

f(x) = ln x, a = 3

{Use of Tech} Approximations with Taylor polynomials

a. Approximate the given quantities using Taylor polynomials with n = 3.

b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.

√1.06

Evaluating an infinite series Let f(x) = (eˣ − 1)/x, for x ≠ 0, and f(0)=1. Use the Taylor series for f centered at 0 to evaluate f(1) and to find the value of ∑ₖ₌₀∞ 1/(k+1)!

{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals

S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt

c. Use the polynomials in part (b) to approximate S(0.05) and C(−0.25).

{Use of Tech} Approximating sin x Let f(x)=sin x, and let pₙ and qₙ be nth−order Taylor polynomials for f centered at 0 and π, respectively.

a. Find p₅ and q₅

b. Graph f, p₅, and q₅ on the interval [−π, 2π]. On what interval is p₅ a better approximation to f than q₅? On what interval is q₅ a better approximation to f than p₅?

c. Complete the following table showing the errors in the approximations given by p₅ and q₅ at selected points.

d. At which points in the table is p₅ a better approximation to f than q₅? At which points do p₅ and q₅ give equal approximations to f? Explain your observations.

Probability: tossing for a head The expected (average) number of tosses of a fair coin required to obtain the first head is ∑ₖ₌₁∞ k(1/2)ᵏ. Evaluate this series and determine the expected number of tosses. (Hint: Differentiate a geometric series.) 

ƒ(x) = eˣ, a = 0; e-0.08

a. Find the Taylor polynomials of order n = 1 and n = 2 for the given functions centered at the given point a.

{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.

sin 0.3, n = 4

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

d. 1/(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

Any method

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients.

b. Determine the radius of convergence of the series.

f(x) = (1 + x²)⁻²/³

Maclaurin Series

Find Taylor series at x = 0 for the functions in Exercises 63–70.

cos (x³/√5)

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

a. Only even powers of x appear in the Taylor polynomials for f(x)=e⁻²ˣ centered at 0.

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.

ln (1 + x²)