15. Power Series

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

Find the interval of convergence for the Taylor series for $f\left(x\right)=\sin x$ centered at $x=\frac{\pi }{2}$.

Find the interval of convergence for the Maclaurin series for $f\left(x\right)={\tan }^{-1}x$

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

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