15. Power Series

Find the Taylor polynomials of order $0,1,2$, and $3$ for $f\left(x\right)=\ln \left(x\right)$ centered at $x=1$.

Approximate $\ln 1.5$ to four decimal places using the third-degree Taylor polynomial for $f\left(x\right)=\ln x$.

Find the Maclaurin polynomials of order $0,1,2$, and $3$ for $f\left(x\right)=\sin x$

Approximate $\sin 0.3$ to four decimal places using the third-degree Maclaurin polynomial for $f\left(x\right)=\sin x$.

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