BC Calculus Series Test

The coefficient of the

 $x^3$  term in the Taylor series for  $e^{2x}$  centered at x = 0 is

• 1/6
  

• 1/3

• 2/3

• 4/3

 $\sum_{n=1}^{\infty}\left(\frac{1}{3}\right)^{2n}$  =

• 1/8

• 1/3

• 1

• 9/8

•  $\infty$  

The coefficient of  $\left(x-1\right)^5$  in the Taylor Series for  $f\left(x\right)=\ln x$  about x = 1 is

• 1/5

• - 1/5

• 1/24

• - 1/24

• 1/120

The series $\sum_{n=0}^{\infty}\left(-9\right)^n\ \frac{x^{4n}}{\left(2n\right)!}$  represents which function below?

•  $\cos\left(3x^2\right)$  

•  $\sin\left(3x^2\right)$  

•  $\cos\left(9x^4\right)$  

•  $\tan^{-1}\left(3x\right)$  

•  $e^{-3x^2}$  

Which of the following series converge?

• 
  

  $\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}$
  

• $\sum_{n=1}^{\infty}\frac{1}{n}$

• $\sum_{n=1}^{\infty}\frac{1}{2n+1}$

• $\sum_{n=1}^{\infty}\frac{1}{n^2+1}$

• $\sum_{n=1}^{\infty}\frac{n}{n^2+1}$

Consider the series $\sum_{n=1}^{\infty}\frac{2^n}{\left(n+1\right)!}$   If the Ratio Test is applied to this series, which of the following inequalities indicates that the series is convergent?

•  $\lim_{n\rightarrow\infty}\ \frac{2}{n+2}<1$  

•  $\lim_{n\rightarrow\infty}\ \frac{n+2}{2}<1$  

•  $\lim_{n\rightarrow\infty}\ \frac{2^n}{n+2}<1$  

•  $\lim_{n\rightarrow\infty}\ \frac{2^n}{\left(n+2\right)!}<1$  

What is the sum of the series

 $\frac{\pi}{e}-\frac{\pi}{e^2}+\frac{\pi}{e^3}-\frac{\pi}{e^4}+...$  

•  $\frac{\pi}{e+\pi}$  

•  $\frac{\pi}{e+1}$  

•  $\frac{\pi}{e-1}$  

• The series diverges

Which of the following series diverge?   $I.\ \ \ \ \sum_{n=0}^{\infty}\left(\frac{e}{\pi}\right)^n$   $II.\ \ \ \ \sum_{n=1}^{\infty}\frac{\sqrt{n}}{n^2}\ \ \$   $III.\ \ \ \ \sum_{n=1}^{\infty}\frac{n!}{3^n}\$  

• III only

• I and II only

• I and III only

• II and III only

• I, II, and II

Let  $P\left(x\right)=3x^2-5x^3+7x^4+3x^5$  be the fifth-degree Taylor Polynomial for the function f about x = 0.  What is the value of  $f'''\left(0\right)$  ?

• -30

• -15

• -5

• - 5/6

• - 1/6

Which of the following is the series for  $f\left(x\right)=x\left(\cos x-1\right)$  

•  $x-\frac{x^3}{2}+\frac{x^5}{24}-...$  

•  $-x^3+x^5-x^7+...$  

•  $\frac{x^3}{2}-\frac{x^5}{24}+\frac{x^7}{720}-...$  

•  $-\frac{x^3}{2}+\frac{x^5}{24}-\frac{x^7}{720}+...$  

•  $1-\frac{x^2}{2}+\frac{x^4}{24}-...$