Power, Taylor, & Maclaurin Series

The power series $$\sum_{n=0}^{\infty\ }a_n\left(x-4\right)^n$$ converges for $$x=7$$ and diverges for $$x=-1$$ . Which of the following must be true? Select all that apply.

The fourth-degree Taylor polynomial for f centered at $$x=-2$$ is given by $$P_4\left(x\right)=1-2\left(x+2\right)+\frac{3}{5}\left(x+2\right)^2+\frac{7}{8}\left(x+2\right)^3-\frac{5}{12}\left(x+2\right)^4$$ . What is $$f^{\left(4\right)}\left(-2\right)$$ ?

Give the first three nonzero terms of the power series for $$f\left(x\right)=\frac{4x}{2-\sqrt[]{x}}$$ .

What is the value of $$\frac{3\pi\ }{4}-\frac{\left(\frac{3\pi\ }{4}\right)^3}{3!}+\frac{\left(\frac{3\pi\ }{4}\right)^5}{5!}-\ldots$$ ?

Which of the following gives the Maclaurin series for $$f\left(x\right)=x^2\sin\left(-x\right)$$ ?

The eighth-degree Taylor polynomial for f centered at $$x=3$$ , $$P_8\left(x\right)$$ , is used to approximate $$f\left(3.5\right)$$ . If $$\left|f^{\left(9\right)}\left(x\right)\right|\le\frac{1}{12}$$ for all x in the interval $$\left[3,3.5\right]$$ , give the Lagrange error bound.

How many nonzero terms must be used in the Maclaurin series for $$\int_0^1e^{-x^2}dx\$$ so that the error in your approximation is no more than $$\frac{1}{200}$$ ?

Find the interval of convergence for $$\sum_{n=1}^{\infty\ }\frac{\left(-1\right)^{n-1}x^n}{n}$$ .

A portion of the graph of $$y=\left|f^{\left(5\right)}\left(x\right)\right|$$ is given. If the fourth-degree Maclaurin polynomial to f is used to approximate $$f\left(1\right)$$ , what is the least possible Lagrange error bound?

What is the fifth nonzero term in the Taylor series for $$f\left(x\right)=\frac{1}{x^2}$$ centered at $$x=2$$ ? Assume the Taylor series is written in the form $$f\left(x\right)=a+bx+cx^2+\ldots$$ .