Taylor Polynomials & Series

What is the sum of the infinite geometric series

 $\frac{3}{2}+\frac{9}{16}+\frac{27}{128}+\frac{81}{1024}+...$  ?

• 1.60

• 2.35

• 2.40

• 2.45

• 2.50

Tags

CCSS.HSF.BF.A.2

CCSS.HSF.LE.A.2

 What is the value of $\sum_{n=1}^{\infty}\frac{2^{n+1}}{3^n}$  ?

• 1

• 2

• 4

• 6

• The series diverges.
  

Tags

CCSS.HSA.SSE.B.4

 If  $f\left(x\right)=\sum_{k=1}^{\infty}\left(\sin^2x\right)^k$  then  $f\left(1\right)$  is?

• 0.369

• 0.585

• 2.400

• 2.426

• 3.426

What is the sum of the series $1+\ln2+\frac{\left(\ln2\right)^2}{2!}+...+\frac{\left(\ln2\right)^n}{n!}+...$  ?

•  $\ln2$  

•  $\ln\left(1+\ln2\right)$  

•  $2$  

•  $e^2$  

• The series diverges.
  

Tags

CCSS.HSF.BF.B.5

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  $f^{'''}\left(0\right)$  ?

•  $-30$  

•  $-15$  

•  $-5$  

•  $-\frac{5}{6}$  

•  $-\frac{1}{6}$  

Let  $f$  be a function with  $f\left(3\right)=2$  , and  $f'\left(3\right)=-1$  ,  $f''\left(3\right)=6$  , and  $f'''\left(3\right)=12$  . Which of the following is the third-degree Taylor polynomial for  $f$  about  $x=3$  ?

•  $2-\left(x-3\right)+3\left(x-3\right)^2+2\left(x-3\right)^3$  

•  $2-\left(x-3\right)+3\left(x-3\right)^2+4\left(x-3\right)^3$  

•  $2-\left(x-3\right)+6\left(x-3\right)^2+12\left(x-3\right)^3$  

•  $2-x+3x^2+2x^3$  

•  $2-x+6x^2+12x^3$  

Tags

CCSS.HSA.APR.A.1

CCSS.HSA.SSE.A.1

CCSS.HSA.SSE.B.3

CCSS.HSF.BF.A.1

Which is the best of the following polynomial approximations to  $\cos2x$  near  $x=0$  ?

•  $1+\frac{x}{2}$  

•  $1+x$  

•  $1-\frac{x^2}{2}$  

•  $1-2x^2$  

•  $1-2x+x^2$  

Tags

CCSS.HSA.SSE.A.2

CCSS.HSA.SSE.B.3

CCSS.HSF.TF.A.2

The third-degree Taylor polynomial about  $x=0$  of  $\ln\left(1-x\right)$  is ?

•  $-x-\frac{x^2}{2}-\frac{x^3}{3}$  

•  $1-x+\frac{x^2}{2}$  

•  $x-\frac{x^2}{2}+\frac{x^3}{3}$  

•  $-1+x-\frac{x^2}{2}$  

•  $-x+\frac{x^2}{2}-\frac{x^3}{3}$  

Tags

CCSS.HSA.SSE.B.3

CCSS.HSF.BF.B.5

The Maclaurin series for  $\frac{1}{1-x}$  is  $\sum_{n=0}^{\infty}x^n$  . Which of the following is a power series expansion for  $\frac{x^2}{1-x^2}$  ?

•  $1+x^2+x^4+x^6+x^8+...$  

•  $x^2+x^3+x^4+x^5+...$  

•  $x^2+2x^3+3x^4+4x^5+...$  

•  $x^2+x^4+x^6+x^8+...$  

•  $x^2-x^4+x^6-x^8+...$  

Tags

CCSS.HSA.APR.D.6

CCSS.HSA.SSE.A.2

CCSS.HSA.SSE.B.3

A series expansion of  $\frac{\sin t}{t}$  is ?

•  $1-\frac{t^2}{3!}+\frac{t^4}{5!}-\frac{t^6}{7!}+...$  

•  $\frac{1}{t}-\frac{t}{2!}+\frac{t^3}{4!}-\frac{t^5}{6!}+...$  

•  $1+\frac{t^2}{3!}+\frac{t^4}{5!}+\frac{t^6}{7!}+...$  

•  $\frac{1}{t}+\frac{t}{2!}+\frac{t^3}{4!}+\frac{t^5}{6!}+...$  

•  $t-\frac{t^3}{3!}+\frac{t^5}{5!}-\frac{t^7}{7!}+...$