15. Power Series

Limits Evaluate the following limits using Taylor series.

lim ₓ→∞ x(e¹/ˣ − 1)

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)