15. Power Series

Limits Evaluate the following limits using Taylor series.

lim ₓ→₀ (tan ⁻¹ x − x)/x³"

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

ƒ(x) = cos⁻¹ x, n = 2, a = 1/2

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

ƒ(x) = sin 2x, n = 3, 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

Limits Evaluate the following limits using Taylor series.

lim ₓ→₀ (eˣ − e⁻ˣ)/x

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

Limits Evaluate the following limits using Taylor series.

lim ₓ→₁ (x 1)/(ln x)

Limits Evaluate the following limits using Taylor series.

lim ₓ→∞ x sin(1/x)

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

Limits Evaluate the following limits using Taylor series.

lim ₓ→₄ (x² 16)/(ln (x 3)}

Limits Evaluate the following limits using Taylor series.

lim ₓ→₀ (sin 2x)/x

Write out the first three terms of the Maclaurin series for the following functions.

ƒ(x) = (1 + x)^(1/3)"

Binomial series Write out the first three terms of the Maclaurin series for the following functions.

ƒ(x) = (1 + 2x)^(-5)

Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.

ƒ(x) = cos x, a = π/2