Question 1

How many coefficients in the binomial series expansion of $( 1 + x ) ^ { 7 }$ are divisible by 7?

Accepted Answer

A) 0 
B) 5 
C) 7 
D) 3 
E) 2
F)6
G)1
H)4 
A) 0 
B) 5 
C) 7 
D) 3 
E) 2
F)6
G)1
H)4

Question 2

Use the binomial series to expand $\frac { 1 } { ( 1 + x ) ^ { 3 } }$ as a power series. State the radius of convergence.

Accepted Answer

The answer of Use the binomial series to expand \(\frac...

Question 3

Determine whether $a _ { n } = \frac { 3 + ( - 1 ) ^ { n } } { n }$ is increasing, decreasing, or not monotonic.

Accepted Answer

The answer of Determine whether \(a _ { n }...

Question 4

Which of the following is the degree 4 Taylor polynomial centered at $a = 0$ for $f ( x ) = \cos ( 2 x )$ ?
1) $1 - ( 2 x ) ^ { 2 } + 16 x ^ { 4 }$ 2) $1 - 2 x ^ { 2 } + \frac { 2 } { 3 } x ^ { 4 }$ 3) $1 - \frac { ( 2 x ) ^ { 2 } } { 2 ! } + \frac { ( 2 x ) ^ { 4 } } { 4 ! }$

Accepted Answer

A) None 
B) 1 
C) 2 
D) 3 
E) 1, 2
F)1, 3
G)2, 3
H)1, 2, 3 
A) None 
B) 1 
C) 2 
D) 3 
E) 1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Question 5

Find the power series for $f ( x ) = \frac { 5 x + 7 } { x ^ { 2 } + 2 x - 3 }$ in terms of powers of x.Hint: Express $f ( x )$ in terms of partial fractions.

Accepted Answer

The answer of Find the power series for \(f (...

Question 6

Given $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 3 } }$ .(a) Approximate the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 3 } }$ by using the sum of the first 4 terms.(b) Estimate the error involved in the approximation in part (a).(c) How many terms are required to ensure that the sum is accurate to within 0:001?

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM&

Question 7

Which one of the following series converges?

Accepted Answer

A)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 3 / 4 } }$ 
B)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$ 
C)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 099 } }$ 
D)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { - 4 } }$ 
E) None of these 
A)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 3 / 4 } }$ 
B)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n }$ 
C)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 099 } }$ 
D)  $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { - 4 } }$ 
E) None of these

Question 8

Which of the following is the degree 3 Taylor polynomial centered at $a = 0$ for $f ( x ) = \cos ( 2 x )$ ?
1) $1 - ( 2 x ) ^ { 2 } + 16 x ^ { 4 }$ 2) $1 - 2 x ^ { 2 } + \frac { 8 } { 3 } x ^ { 4 }$ 3) $1 - \frac { ( 2 x ) ^ { 2 } } { 2 ! }$

Accepted Answer

A) None 
B) 1 
C) 2 
D) 3 
E) 1, 2
F)1, 3
G)2, 3
H)1, 2, 3 
A) None 
B) 1 
C) 2 
D) 3 
E) 1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Question 9

Find the radius of convergence of $\sum _ { n = 1 } ^ { \infty } \frac { n } { 2 \cdot 5 \cdot 8 \cdots \cdot ( 3 n - 1 ) } x ^ { n }$ .

Accepted Answer

The answer of Find the radius of convergence of \(\sum...

Question 10

Find the interval of convergence of $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x + 2 ) ^ { n } } { \sqrt { n } 3 ^ { n } }$ .

Accepted Answer

A)  $( - \infty , \infty )$ 
B)  $( - 5,1 ]$ 
C)  $[ - 5,1 )$ 
D)  $[ - 5,1 ]$ 
E)  $( - 4,0 ]$ 
F) $[ - 4,0 )$ 
G) $[ - 4,0 ]$ 
H) $( - 3 , - 1 )$ 
A)  $( - \infty , \infty )$ 
B)  $( - 5,1 ]$ 
C)  $[ - 5,1 )$ 
D)  $[ - 5,1 ]$ 
E)  $( - 4,0 ]$ 
F) $[ - 4,0 )$ 
G) $[ - 4,0 ]$ 
H) $( - 3 , - 1 )$

Question 11

Determine whether the series $\sum _ { n = 0 } ^ { \infty } \frac { 1 + \sin ^ { 2 } n } { 5 ^ { n } }$ converges.

Accepted Answer

The answer of Determine whether the series \(\sum _ {...

Question 12

Which of the following series can be shown to be convergent using the Ratio Test?
1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$ 
2) $\sum _ { n = 1 } ^ { \infty } \frac { n } { 3 ^ { n } }$ 
3) $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } } { n ! }$

Accepted Answer

A) None 
B) 1 
C) 2 
D) 3 
E) 1, 2
F)1, 3
G)2, 3
H)1, 2, 3 
A) None 
B) 1 
C) 2 
D) 3 
E) 1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Question 13

Determine if the series $\sum _ { n = 2 } ^ { \infty } \left( \frac { n } { \ln n } \right) ^ { n }$ converges or diverges by the Ratio Test or Root Test.

Accepted Answer

The answer of Determine if the series \(\sum _ {...

Question 14

Consider the recursive sequence defined by $a _ { 1 } = 2 ; a _ { n + 1 } = \frac { 2 } { 3 - a _ { n } } , n > 1$ .(a) Evaluate the first four terms of this sequence.(b) Show that the sequence converges.(c) Find the limit.

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM&

Question 15

Find a formula for the general term $a _ { n }$ of the sequence $\left\{ 2,1 , \frac { 8 } { 9 } , 1 , \frac { 32 } { 25 } , \frac { 64 } { 36 } , \frac { 128 } { 49 } , \ldots \right\}$ .

Accepted Answer

A)  $\frac { 2 ^ { n + 1 } } { n ^ { 2 } }$ 
B)  $\frac { 2 ^ { n } } { ( n + 1 ) ^ { 2 } }$ 
C)  $\frac { 2 ^ { n } } { n ^ { 2 } }$ 
D)  $\frac { 2 ^ { n } } { ( n - 1 ) ^ { 2 } }$ 
E)  $\frac { 2 ^ { n - 1 } } { n ^ { 2 } }$
F) $\frac { 2 n } { n ^ { 2 } }$ 
G) $\frac { 2 ^ { n - 1 } } { n ^ { 2 } }$ 
H)None of these 
A)  $\frac { 2 ^ { n + 1 } } { n ^ { 2 } }$ 
B)  $\frac { 2 ^ { n } } { ( n + 1 ) ^ { 2 } }$ 
C)  $\frac { 2 ^ { n } } { n ^ { 2 } }$ 
D)  $\frac { 2 ^ { n } } { ( n - 1 ) ^ { 2 } }$ 
E)  $\frac { 2 ^ { n - 1 } } { n ^ { 2 } }$
F) $\frac { 2 n } { n ^ { 2 } }$ 
G) $\frac { 2 ^ { n - 1 } } { n ^ { 2 } }$ 
H)None of these

Question 16

How many terms of the alternating series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } n ^ { - 2 }$ must we add in order to be sure that the partial sum $S _ { n }$ is within 0.0001 of the sum $S$ ?

Accepted Answer

A) 10 
B) 300 
C) 30,000 
D) 30 
E) 3
F)10,000
G)100
H)1000 
A) 10 
B) 300 
C) 30,000 
D) 30 
E) 3
F)10,000
G)100
H)1000

Question 17

Determine whether the series $\sum _ { n = 1 } ^ { \infty } ( 1.0001 ) ^ { n }$ is convergent or divergent. If it is convergent, find the sum.

Accepted Answer

The answer of Determine whether the series \(\sum _ {...

Question 18

Find the radius of convergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { n } ( x + 1 ) ^ { n } } { 3 ^ { n } }$ .

Accepted Answer

A) 0 
B) 2 
C) 3 
D) 4 
E)  $\frac { 1 } { 2 }$ 
F)1
G)5
H) $\infty$ 
A) 0 
B) 2 
C) 3 
D) 4 
E)  $\frac { 1 } { 2 }$ 
F)1
G)5
H) $\infty$

Question 19

Which of the following series will, when rearranged, converge to different values?
1) $\sum _ { n = 1 } ^ { \infty } n ^ { - 1 }$ 
2) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } n ^ { - 1 }$ 
3) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } n ^ { - 2 }$

Accepted Answer

A) None 
B) 1 
C) 2 
D) 3 
E) 1, 2
F)1, 3
G)2, 3
H)1, 2, 3 
A) None 
B) 1 
C) 2 
D) 3 
E) 1, 2
F)1, 3
G)2, 3
H)1, 2, 3

Question 20

Find the interval of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { ( - 3 x ) ^ { n } } { 3 n + 1 }$ .

Accepted Answer

A)  $\left( - \frac { 1 } { 3 } , \frac { 1 } { 3 } \right)$ 
B)  $\left( - \frac { 1 } { 3 } , \frac { 1 } { 3 } \right]$ 
C)  $\left[ - \frac { 1 } { 3 } , \frac { 1 } { 3 } \right)$ 
D)  $\left[ - \frac { 1 } { 3 } , \frac { 1 } { 3 } \right]$ 
E)  $( - 3,3 )$ 
F) $( - 3,3 ]$ 
G) $[ - 3,3 )$ 
H) $[ - 3,3 ]$ 
A)  $\left( - \frac { 1 } { 3 } , \frac { 1 } { 3 } \right)$ 
B)  $\left( - \frac { 1 } { 3 } , \frac { 1 } { 3 } \right]$ 
C)  $\left[ - \frac { 1 } { 3 } , \frac { 1 } { 3 } \right)$ 
D)  $\left[ - \frac { 1 } { 3 } , \frac { 1 } { 3 } \right]$ 
E)  $( - 3,3 )$ 
F) $( - 3,3 ]$ 
G) $[ - 3,3 )$ 
H) $[ - 3,3 ]$

How many coefficients in the binomial series expansion of $( 1 + x ) ^ { 7 }$ are divisible by 7?

Use the binomial series to expand $\frac { 1 } { ( 1 + x ) ^ { 3 } }$ as a power series. State the radius of convergence.

Determine whether $a _ { n } = \frac { 3 + ( - 1 ) ^ { n } } { n }$ is increasing, decreasing, or not monotonic.

Which of the following is the degree 4 Taylor polynomial centered at $a = 0$ for $f ( x ) = \cos ( 2 x )$ ? 1) $1 - ( 2 x ) ^ { 2 } + 16 x ^ { 4 }$ 2) $1 - 2 x ^ { 2 } + \frac { 2 } { 3 } x ^ { 4 }$ 3) $1 - \frac { ( 2 x ) ^ { 2 } } { 2 ! } + \frac { ( 2 x ) ^ { 4 } } { 4 ! }$

Find the power series for $f ( x ) = \frac { 5 x + 7 } { x ^ { 2 } + 2 x - 3 }$ in terms of powers of x.Hint: Express $f ( x )$ in terms of partial fractions.

Which one of the following series converges?

Which of the following is the degree 3 Taylor polynomial centered at $a = 0$ for $f ( x ) = \cos ( 2 x )$ ? 1) $1 - ( 2 x ) ^ { 2 } + 16 x ^ { 4 }$ 2) $1 - 2 x ^ { 2 } + \frac { 8 } { 3 } x ^ { 4 }$ 3) $1 - \frac { ( 2 x ) ^ { 2 } } { 2 ! }$

Find the radius of convergence of $\sum _ { n = 1 } ^ { \infty } \frac { n } { 2 \cdot 5 \cdot 8 \cdots \cdot ( 3 n - 1 ) } x ^ { n }$ .

Find the interval of convergence of $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x + 2 ) ^ { n } } { \sqrt { n } 3 ^ { n } }$ .

Determine whether the series $\sum _ { n = 0 } ^ { \infty } \frac { 1 + \sin ^ { 2 } n } { 5 ^ { n } }$ converges.

Which of the following series can be shown to be convergent using the Ratio Test? 1) $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } }$ 2) $\sum _ { n = 1 } ^ { \infty } \frac { n } { 3 ^ { n } }$ 3) $\sum _ { n = 1 } ^ { \infty } \frac { 2 ^ { n } } { n ! }$

Determine if the series $\sum _ { n = 2 } ^ { \infty } \left( \frac { n } { \ln n } \right) ^ { n }$ converges or diverges by the Ratio Test or Root Test.

Consider the recursive sequence defined by $a _ { 1 } = 2 ; a _ { n + 1 } = \frac { 2 } { 3 - a _ { n } } , n > 1$ .(a) Evaluate the first four terms of this sequence.(b) Show that the sequence converges.(c) Find the limit.

Find a formula for the general term $a _ { n }$ of the sequence $\left\{ 2,1 , \frac { 8 } { 9 } , 1 , \frac { 32 } { 25 } , \frac { 64 } { 36 } , \frac { 128 } { 49 } , \ldots \right\}$ .

How many terms of the alternating series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } n ^ { - 2 }$ must we add in order to be sure that the partial sum $S _ { n }$ is within 0.0001 of the sum $S$ ?

Determine whether the series $\sum _ { n = 1 } ^ { \infty } ( 1.0001 ) ^ { n }$ is convergent or divergent. If it is convergent, find the sum.

Find the radius of convergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { n } ( x + 1 ) ^ { n } } { 3 ^ { n } }$ .

Which of the following series will, when rearranged, converge to different values? 1) $\sum _ { n = 1 } ^ { \infty } n ^ { - 1 }$ 2) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } n ^ { - 1 }$ 3) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } n ^ { - 2 }$

Find the interval of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { ( - 3 x ) ^ { n } } { 3 n + 1 }$ .

Functions and Models

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Integrals

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Differential Equations

Vectors and the Geometry of Space

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Multiple Integrals

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Exam 8: Infinite Sequences and Series

How many coefficients in the binomial series expansion of (1+x)7( 1 + x ) ^ { 7 }(1+x)7 are divisible by 7?

Use the binomial series to expand 1(1+x)3\frac { 1 } { ( 1 + x ) ^ { 3 } }(1+x)31​ as a power series. State the radius of convergence.

Determine whether an=3+(−1)nna _ { n } = \frac { 3 + ( - 1 ) ^ { n } } { n }an​=n3+(−1)n​ is increasing, decreasing, or not monotonic.

Find the power series for f(x)=5x+7x2+2x−3f ( x ) = \frac { 5 x + 7 } { x ^ { 2 } + 2 x - 3 }f(x)=x2+2x−35x+7​ in terms of powers of x.Hint: Express f(x)f ( x )f(x) in terms of partial fractions.

Which one of the following series converges?

Find the radius of convergence of ∑n=1∞n2⋅5⋅8⋯⋅(3n−1)xn\sum _ { n = 1 } ^ { \infty } \frac { n } { 2 \cdot 5 \cdot 8 \cdots \cdot ( 3 n - 1 ) } x ^ { n }∑n=1∞​2⋅5⋅8⋯⋅(3n−1)n​xn .

Find the interval of convergence of ∑n=1∞(−1)n(x+2)nn3n\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x + 2 ) ^ { n } } { \sqrt { n } 3 ^ { n } }∑n=1∞​n​3n(−1)n(x+2)n​ .

Determine whether the series ∑n=0∞1+sin⁡2n5n\sum _ { n = 0 } ^ { \infty } \frac { 1 + \sin ^ { 2 } n } { 5 ^ { n } }∑n=0∞​5n1+sin2n​ converges.

Determine if the series ∑n=2∞(nln⁡n)n\sum _ { n = 2 } ^ { \infty } \left( \frac { n } { \ln n } \right) ^ { n }∑n=2∞​(lnnn​)n converges or diverges by the Ratio Test or Root Test.

Consider the recursive sequence defined by a1=2;an+1=23−an,n>1a _ { 1 } = 2 ; a _ { n + 1 } = \frac { 2 } { 3 - a _ { n } } , n > 1a1​=2;an+1​=3−an​2​,n>1 .(a) Evaluate the first four terms of this sequence.(b) Show that the sequence converges.(c) Find the limit.

Find a formula for the general term ana _ { n }an​ of the sequence {2,1,89,1,3225,6436,12849,…}\left\{ 2,1 , \frac { 8 } { 9 } , 1 , \frac { 32 } { 25 } , \frac { 64 } { 36 } , \frac { 128 } { 49 } , \ldots \right\}{2,1,98​,1,2532​,3664​,49128​,…} .

How many terms of the alternating series ∑n=1∞(−1)n+1n−2\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } n ^ { - 2 }∑n=1∞​(−1)n+1n−2 must we add in order to be sure that the partial sum SnS _ { n }Sn​ is within 0.0001 of the sum SSS ?

Determine whether the series ∑n=1∞(1.0001)n\sum _ { n = 1 } ^ { \infty } ( 1.0001 ) ^ { n }∑n=1∞​(1.0001)n is convergent or divergent. If it is convergent, find the sum.

Find the radius of convergence of the series ∑n=1∞nn(x+1)n3n\sum _ { n = 1 } ^ { \infty } \frac { n ^ { n } ( x + 1 ) ^ { n } } { 3 ^ { n } }∑n=1∞​3nnn(x+1)n​ .

Find the interval of convergence of ∑n=0∞(−3x)n3n+1\sum _ { n = 0 } ^ { \infty } \frac { ( - 3 x ) ^ { n } } { 3 n + 1 }∑n=0∞​3n+1(−3x)n​ .

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Differential Equations

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

How many coefficients in the binomial series expansion of $( 1 + x ) ^ { 7 }$ are divisible by 7?

Use the binomial series to expand $\frac { 1 } { ( 1 + x ) ^ { 3 } }$ as a power series. State the radius of convergence.

Determine whether $a _ { n } = \frac { 3 + ( - 1 ) ^ { n } } { n }$ is increasing, decreasing, or not monotonic.

Find the power series for $f ( x ) = \frac { 5 x + 7 } { x ^ { 2 } + 2 x - 3 }$ in terms of powers of x.Hint: Express $f ( x )$ in terms of partial fractions.

Find the radius of convergence of $\sum _ { n = 1 } ^ { \infty } \frac { n } { 2 \cdot 5 \cdot 8 \cdots \cdot ( 3 n - 1 ) } x ^ { n }$ .

Find the interval of convergence of $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( x + 2 ) ^ { n } } { \sqrt { n } 3 ^ { n } }$ .

Determine whether the series $\sum _ { n = 0 } ^ { \infty } \frac { 1 + \sin ^ { 2 } n } { 5 ^ { n } }$ converges.

Determine if the series $\sum _ { n = 2 } ^ { \infty } \left( \frac { n } { \ln n } \right) ^ { n }$ converges or diverges by the Ratio Test or Root Test.

Consider the recursive sequence defined by $a _ { 1 } = 2 ; a _ { n + 1 } = \frac { 2 } { 3 - a _ { n } } , n > 1$ .(a) Evaluate the first four terms of this sequence.(b) Show that the sequence converges.(c) Find the limit.

Find a formula for the general term $a _ { n }$ of the sequence $\left\{ 2,1 , \frac { 8 } { 9 } , 1 , \frac { 32 } { 25 } , \frac { 64 } { 36 } , \frac { 128 } { 49 } , \ldots \right\}$ .

How many terms of the alternating series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n + 1 } n ^ { - 2 }$ must we add in order to be sure that the partial sum $S _ { n }$ is within 0.0001 of the sum $S$ ?

Determine whether the series $\sum _ { n = 1 } ^ { \infty } ( 1.0001 ) ^ { n }$ is convergent or divergent. If it is convergent, find the sum.

Find the radius of convergence of the series $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { n } ( x + 1 ) ^ { n } } { 3 ^ { n } }$ .

Find the interval of convergence of $\sum _ { n = 0 } ^ { \infty } \frac { ( - 3 x ) ^ { n } } { 3 n + 1 }$ .