Series Representations of Functions (College Board AP® Calculus BC): Exam Questions

What is the approximation of the value of $\cos \left(-1\right)$ found by using the fourth-degree Taylor polynomial for $\cos x$ about $x=0$?

The table above gives selected values for a function $f$ and its first three derivatives. Which of the following is the third-degree Taylor polynomial for $f$ about $x=4$?

The graph of the function represented by the Maclaurin series $1-\frac{x^4}{2}+\frac{x^8}{24}-\frac{x^{12}}{720}+...+\frac{{\left(-1\right)}^n{x}^{4n}}{\left(2n\right)!}+...$ intersects the graph of $y=x^3$ at $x=$

The coefficient of $x^3$ in the Taylor series for $e^{-4x}$ about $x=0$ is

A function $f$ is given in power series form as $f\left(x\right)=\sum _{n=0}^{\infty }{a}_n{x}^n$, which is known to converge for all real values of $x$. $f^{\text{'}}\left(2\right)=$

What are all values of $x$ for which the series $\sum _{n=1}^{\infty }\frac{2x^n}{n}$ converges?

Which of the following is an approximation for the value of $\pi$ that can be found by using the third-degree Taylor polynomial for $\arctan x$ about $x=0$?

Let $f$ be the function defined by $f\left(x\right)=\ln \left(5-x\right)$. The third-degree Taylor polynomial for $f$ about $x=4$ is

The function $f$ is defined for $x>0$ by $f\left(x\right)=\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n{x}^{2n}}{\left(2n+1\right)!}$. What is $\underset{x\to 2}{\lim }f\left(x\right)$?

The coefficient of $x^3$ in the Taylor series for $\frac{\cos \left(3x\right)}{e^x}$ about $x=0$ is

Let $f$ be a function such that $f^{\text{'}}\left(x\right)=\cos \left(2x^2\right)$. The coefficient of $x^9$ in the Taylor series for $f\left(x\right)$ about $x=0$ is

What are all values of $x$ for which the series $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^n{\left(x+3\right)}^n}{n·4^n}$ converges?

The Taylor series for a function $f$ about $x=3$ is given by $f\left(x\right)=\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^n{\left(x-3\right)}^n}{2^n·\left(n^2-n+3\right)}$, which converges for $1\le x\le 5$. Let $P_n\left(x\right)$ be the nth-degree Taylor polynomial for $f$ about $x=3$. Of the following, which is the smallest number $M$ for which the alternating series error bound guarantees that $\left|f\left(x\right)-P_4\left(x\right)\right|\le M$ for all $x$ in the interval $\frac{7}{2}\le x\le \frac{9}{2}$?