Taylor Series

The graph of the function f(x) is shown. Which of the following could be a portion of the graph of the Maclaurin polynomial of degree 13 for f(x)?

$$-1+6\left(x-1\right)-24\left(x-1\right)^2+80\left(x-1\right)^3.$$  

This is a third-degree Taylor polynomial for a function f about x = 1. What is the value of  $$f'''\left(1\right)$$   ?

$$\sum_{n=0}^{\infty}\left(-1\right)^n\frac{\pi^{2n}}{\left(2n\right)!}=$$  

Which of the following is the fourth-degree Taylor Polynomial for $$f\left(x\right)=x\sin x$$  about  $$x=0$$  .

$$\sum_{n=1}^{\infty}a_n\left(x-3\right)^n$$  

The power series shown above converges conditionally at x = 7. Which of the following statements about the convergence of the series at x = -2 is true?

The coefficient of $$\left(x-\frac{\pi}{4}\right)^3$$  in the Taylor series about $$\frac{\pi}{4}$$  of $$f\left(x\right)=\cos x$$  is

Let $$f$$ be the function defined by $$f\left(x\right)=\ln\left(x+4\right)$$  .  The third-degree Taylor Polynomial for $$f$$  centered about $$x=-3$$ is

The radius of convergence of the series $$\sum_{n\ =\ 1}^{\infty}\frac{x^n}{n^3}$$  is 1. If the function f is defined by $$f\left(x\right)=\sum_{n\ =\ 1}^{\infty}\frac{x^n}{n^3},$$  then the interval of convergence for the series representing the derivative of f(x) is

Let  $$f$$  be the function that has derivatives of all orders for all real numbers.  Assume  $$f\left(0\right)=6,\ f'\left(0\right)=-5,\ f''\left(0\right)=14,\ and\ f'''\left(0\right)=36$$  Write a third-order Taylor Polynomial for  $$f$$  at  $$x=0$$   and use it to approximate  $$f\left(0.5\right)$$  .

The coefficient of the   $$x^2-$$ term in the Taylor Polynomial for  $$y=x^{\frac{2}{3}}$$  around  $$x=8$$  is