15. Power Series

Taylor series and interval of convergence

c. Determine the interval of convergence of the series.

f(x) = cosh 3x, 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) = 1/x, a = 1

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Give the interval of convergence for the new series (Theorem 11.4 is useful). Use the Maclaurin series

√(1 + x) = 1 + x/2 − x²/8 + x³/16 − ⋯, −1 ≤ x ≤ 1.

√(9 − 9x)

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) = ln (x − 2), a = 3

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) = e²ˣ, a = 0

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) − 3y = 10, y(0) = 2

Use series to evaluate the limits in Exercises 29–40.

37. lim (x → 0) ln(1 + x²) / (1 - cos(x))

The first three Taylor polynomials for f(x)=√(1+x) centered at 0 are p₀ = 1, p₁ = 1+x/2, and p₂ = 1 + x/2 − x²/8. Find three approximations to √1.1.

{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.

sin x ≈ x − x³/6 on [π/4, π/4]

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) = 1/(1 + 5x)

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.

{(eˣ−1)/x if x ≠ 1, 1 if x = 1

Find the Taylor Series of $f\left(x\right)=\cos x$ centered $x=\pi$. Then, write the power series using summation notation.

Taylor series

b. Write the power series using summation notation.

f(x) = 2ˣ, a = 1

{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.

∫₀⁰ᐧ³⁵ tan ⁻¹x dx

A limit by Taylor series Use Taylor series to evaluate lim ₓ→₀ ((sin x)/x)¹/ˣ²