BackConvergence Write the remainder term Rₙ(x) for the Taylor series for the following functions centered at the given point a. Then show that lim ₙ → ∞ |Rₙ(x)| = 0, for all x in the given interval.
ƒ(x) = sinh x + cosh x, a = 0, - ∞ < x < ∞
Approximate to four decimal places using the third-degree Taylor polynomial for .
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.