15. Power Series

Taylor series and interval of convergence

b. Write the power series using summation notation.

f(x) = tan ⁻¹ (x/2), a = 0

Limits Evaluate the following limits using Taylor series.

lim ₓ→₀ (eˣ − 1)/x

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

ƒ(x) = cosh x, n = 3, a = ln 2

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

ƒ(x) = sinh (-3x), n = 3, a = 0

Taylor series

a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.

f(x)=sin x, a = π/2

Power series from the geometric series Use the geometric series a Σₖ ₌ ₀ ∞ (x)ᵏ = 1/(1 - x), for |x| < 1, to determine the Maclaurin series and the interval of convergence for the following functions.

ƒ(x) = ln (1 - 4x)

{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

e. How many terms of the Maclaurin series are required to approximate C(−0.25) with an error no greater than 10⁻⁶?

{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.

tan x ≈ x on [−π/6, π/6]

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.

∑ₖ₌₁∞ (3x + 2)ᵏ/k

Taylor series

b. Write the power series using summation notation.

f(x) = 1/x, 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 = π/2

{Use of Tech} Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.

f(x) =∛x with a=64; approximate ∛60.

Taylor series

a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.

f(x) = ln x, a = 3

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.

sinh x²

Differential equations

a. Find a power series for the solution of the following differential equations, subject to the given initial condition

b. Identify the function represented by the power series.

y′(t) − y = 0, y(0) = 2