91Ó°ÊÓ

Determine the radius and interval of convergence of the following power series. $$\sum_{k=0}^{\infty} \frac{(2 x)^{k}}{k !}$$

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. b. Write the power series using summation notation. $$f(x)=3^{x}, a=0$$

Find the Taylor polynomials \(p_{1}, p_{2},\) and \(p_{3}\) centered at \(a=1\) for \(f(x)=x^{3}\).

Determine the radius and interval of convergence of the following power series. $$-\frac{x^{2}}{1 !}+\frac{x^{4}}{2 !}-\frac{x^{6}}{3 !}+\frac{x^{8}}{4 !}-\cdots$$

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. b. Write the power series using summation notation. $$f(x)=\log _{3}(x+1), a=0$$

Find the Taylor polynomials \(p_{3}\) and \(p_{4}\) centered at \(a=1\) for \(f(x)=8 \sqrt{x}\).

Determine the radius and interval of convergence of the following power series. $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}(x-1)^{k}}{k}$$

$$\text { Limits Evaluate the following limits using Taylor series.}$$ $$\lim _{x \rightarrow \infty} x\left(e^{1 / x}-1\right)$$

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. b. Write the power series using summation notation. $$f(x)=\cosh 3 x, a=0$$

$$\text { Limits Evaluate the following limits using Taylor series.}$$ $$\lim _{x \rightarrow 0} \frac{e^{-2 x}-4 e^{-x / 2}+3}{2 x^{2}}$$