Question 1

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } + 3 }$

Accepted Answer

The answer of Determine whether the series is convergent or...

Question 2

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\sqrt [ 4 ] { 1 + x ^ { 6 } }$

Accepted Answer

The answer of Use the binomial series to expand the...

Question 3

Determine whether the given series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n } + 3 ^ { n } } { 9 ^ { n } }$

Accepted Answer

The answer of Determine whether the given series converges or...

Question 4

Test the series for convergence or divergence. $\sum _ { n = 1 } ^ { \infty } \sin \left( \frac { 1 } { n ^ { 4 } } \right)$

Accepted Answer

The answer of Test the series for convergence or divergence....

Question 5

Find the value of the limit for the sequence. $\left\{ \arctan \left( \frac { 6 n } { 6 n + 8 } \right) \right\}$

Accepted Answer

The answer of Find the value of the limit for...

Question 6

Evaluate the indefinite integral as an infinite series. $\int \frac { 5 \left( e ^ { x } - 1 \right) } { x } d x$

Accepted Answer

The answer of Evaluate the indefinite integral as an infinite...

Question 7

Find an approximation of the sum of the series accurate to two decimal places. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ^ { 4 } }$

Accepted Answer

A)  -0.95 
B)  -1.02 
C)  -0.87 
D)  -1.09 
A)  -0.95 
B)  -1.02 
C)  -0.87 
D)  -1.09

Question 8

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } - 6 n + 14 }$

Accepted Answer

A)  converges 
B)  diverges 
A)  converges 
B)  diverges

Question 9

Use series to evaluate the limit correct to three decimal places. $\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }$ Select the correct answer.

Accepted Answer

A)  118.933 
B)  114.133 
C)  34.3233 
D)  114.333 
E)  115.933 
A)  118.933 
B)  114.133 
C)  34.3233 
D)  114.333 
E)  115.933

Question 10

Determine whether the series converges or diverges. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } n } { 2 ^ { n } }$

Accepted Answer

The answer of Determine whether the series converges or diverges....

Question 11

Evaluate the function $f ( x ) = \cos x$ by a Taylor polynomial of degree $4$ centered at $a = 0$ , and $x = \frac { \pi } { 4 }$ .

Accepted Answer

A)  $0.7074$ 
B)  $4.2074$ 
C)  $3.2074$ 
D)  $2.2074$ 
E)  $1.2074$ 
A)  $0.7074$ 
B)  $4.2074$ 
C)  $3.2074$ 
D)  $2.2074$ 
E)  $1.2074$

Question 12

Use the Alternating Series Estimation Theorem or Taylor's Inequality to estimate the range of values of x for which the given approximation is accurate to within the stated error. $\cos x \approx 1 - \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 4 } } { 24 } \quad \mid \text { error } \mid < 0.08$ Write a such that $- a < x < a$ .

Accepted Answer

The answer of Use the Alternating Series Estimation Theorem or...

Question 13

Determine whether the sequence defined as follows is convergent or divergent. $a _ { 1 } = 1 , a _ { n + 1 } = 6 - a _ { n } \text { for } n \geq 1$

Accepted Answer

The answer of Determine whether the sequence defined as follows...

Question 14

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 6 } } { ( 5 n ) ! }$

Accepted Answer

The answer of Determine whether the series is convergent or...

Question 15

Use the Comparison Test to determine whether the series is convergent or divergent. $\sum _ { n = 3 } ^ { \infty } \frac { 7 ^ { n } } { 2 ^ { n } - 3 }$

Accepted Answer

The answer of Use the Comparison Test to determine whether...

Question 16

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\frac { 1 } { ( 4 + x ) ^ { 5 } }$

Accepted Answer

The answer of Use the binomial series to expand the...

Question 17

Find the value of the limit of the sequence defined by $a _ { 1 } = 1 , a _ { n + 1 } = 6 - \frac { 1 } { a _ { n } }$ .

Accepted Answer

A)  $\frac { 6 + \sqrt { 5 } } { 2 }$ 
B)  $\frac { 6 - \sqrt { 10 } } { 2 }$ 
C)  $\frac { 6 + \sqrt { 10 } } { 2 }$ 
D)  $\frac { 6 - \sqrt { 5 } } { 2 }$ 
E)  $6 + \sqrt { 10 }$ 
A)  $\frac { 6 + \sqrt { 5 } } { 2 }$ 
B)  $\frac { 6 - \sqrt { 10 } } { 2 }$ 
C)  $\frac { 6 + \sqrt { 10 } } { 2 }$ 
D)  $\frac { 6 - \sqrt { 5 } } { 2 }$ 
E)  $6 + \sqrt { 10 }$

Question 18

Determine whether the given series converges or diverges. If it converges, find its sum. $\sum _ { n = 1 } ^ { \infty } \frac { 8 ^ { n } + 7 } { 8 ^ { n + 1 } }$

Accepted Answer

The answer of Determine whether the given series converges or...

Question 19

Find an expression for the $n ^ { \text {th } }$ term of the sequence. (Assume that the pattern continues.) $\left\{ \frac { 2 } { 25 } , \frac { 4 } { 36 } , \frac { 6 } { 49 } , \frac { 8 } { 64 } , \frac { 10 } { 81 } , \cdots \right\}$

Accepted Answer

The answer of Find an expression for the \(n ^...

Question 20

Find the partial sum $S _ { 7 }$ of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 3 + 4 ^ { n } }$ . Give your answer to five decimal places.

Accepted Answer

A)  $S _ { 7 } = 0.18975$ 
B)  $S _ { 7 } = 0.19176$ 
C)  $S _ { 7 } = 0.18985$ 
D)  $S _ { 7 } = 0.21555$ 
E)  $S _ { 7 } = 1.60976$ 
A)  $S _ { 7 } = 0.18975$ 
B)  $S _ { 7 } = 0.19176$ 
C)  $S _ { 7 } = 0.18985$ 
D)  $S _ { 7 } = 0.21555$ 
E)  $S _ { 7 } = 1.60976$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } + 3 }$

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\sqrt [ 4 ] { 1 + x ^ { 6 } }$

Determine whether the given series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n } + 3 ^ { n } } { 9 ^ { n } }$

Test the series for convergence or divergence. $\sum _ { n = 1 } ^ { \infty } \sin \left( \frac { 1 } { n ^ { 4 } } \right)$

Find the value of the limit for the sequence. $\left\{ \arctan \left( \frac { 6 n } { 6 n + 8 } \right) \right\}$

Evaluate the indefinite integral as an infinite series. $\int \frac { 5 \left( e ^ { x } - 1 \right) } { x } d x$

Find an approximation of the sum of the series accurate to two decimal places. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ^ { 4 } }$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } - 6 n + 14 }$

Use series to evaluate the limit correct to three decimal places. $\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }$ Select the correct answer.

Determine whether the series converges or diverges. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } n } { 2 ^ { n } }$

Evaluate the function $f ( x ) = \cos x$ by a Taylor polynomial of degree $4$ centered at $a = 0$ , and $x = \frac { \pi } { 4 }$ .

Determine whether the sequence defined as follows is convergent or divergent. $a _ { 1 } = 1 , a _ { n + 1 } = 6 - a _ { n } \text { for } n \geq 1$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 6 } } { ( 5 n ) ! }$

Use the Comparison Test to determine whether the series is convergent or divergent. $\sum _ { n = 3 } ^ { \infty } \frac { 7 ^ { n } } { 2 ^ { n } - 3 }$

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\frac { 1 } { ( 4 + x ) ^ { 5 } }$

Find the value of the limit of the sequence defined by $a _ { 1 } = 1 , a _ { n + 1 } = 6 - \frac { 1 } { a _ { n } }$ .

Determine whether the given series converges or diverges. If it converges, find its sum. $\sum _ { n = 1 } ^ { \infty } \frac { 8 ^ { n } + 7 } { 8 ^ { n + 1 } }$

Find an expression for the $n ^ { \text {th } }$ term of the sequence. (Assume that the pattern continues.) $\left\{ \frac { 2 } { 25 } , \frac { 4 } { 36 } , \frac { 6 } { 49 } , \frac { 8 } { 64 } , \frac { 10 } { 81 } , \cdots \right\}$

Find the partial sum $S _ { 7 }$ of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 3 + 4 ^ { n } }$ . Give your answer to five decimal places.

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Exam 11: Infinite Sequences and Series

Determine whether the series is convergent or divergent. ∑n=1∞1n2+3\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } + 3 }∑n=1∞​n2+31​

Use the binomial series to expand the function as a power series. Find the radius of convergence. 1+x64\sqrt [ 4 ] { 1 + x ^ { 6 } }41+x6​

Determine whether the given series converges or diverges. If it converges, find its sum. ∑n=0∞7n+3n9n\sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n } + 3 ^ { n } } { 9 ^ { n } }∑n=0∞​9n7n+3n​

Test the series for convergence or divergence. ∑n=1∞sin⁡(1n4)\sum _ { n = 1 } ^ { \infty } \sin \left( \frac { 1 } { n ^ { 4 } } \right)∑n=1∞​sin(n41​)

Find the value of the limit for the sequence. {arctan⁡(6n6n+8)}\left\{ \arctan \left( \frac { 6 n } { 6 n + 8 } \right) \right\}{arctan(6n+86n​)}

Evaluate the indefinite integral as an infinite series. ∫5(ex−1)xdx\int \frac { 5 \left( e ^ { x } - 1 \right) } { x } d x∫x5(ex−1)​dx

Find an approximation of the sum of the series accurate to two decimal places. ∑n=1∞(−1)nn4\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ^ { 4 } }∑n=1∞​n4(−1)n​

Determine whether the series is convergent or divergent. ∑n=1∞1n2−6n+14\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } - 6 n + 14 }∑n=1∞​n2−6n+141​

Use series to evaluate the limit correct to three decimal places. lim⁡x→07x−tan⁡−17xx3\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }limx→0​x37x−tan−17x​ Select the correct answer.

Determine whether the series converges or diverges. ∑n=1∞(−1)nn2n\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } n } { 2 ^ { n } }∑n=1∞​2n(−1)nn​

Evaluate the function f(x)=cos⁡xf ( x ) = \cos xf(x)=cosx by a Taylor polynomial of degree 444 centered at a=0a = 0a=0 , and x=π4x = \frac { \pi } { 4 }x=4π​ .

Determine whether the sequence defined as follows is convergent or divergent. a1=1,an+1=6−an for n≥1a _ { 1 } = 1 , a _ { n + 1 } = 6 - a _ { n } \text { for } n \geq 1a1​=1,an+1​=6−an​ for n≥1

Determine whether the series is convergent or divergent. ∑n=1∞(n!)6(5n)!\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 6 } } { ( 5 n ) ! }∑n=1∞​(5n)!(n!)6​

Use the Comparison Test to determine whether the series is convergent or divergent. ∑n=3∞7n2n−3\sum _ { n = 3 } ^ { \infty } \frac { 7 ^ { n } } { 2 ^ { n } - 3 }∑n=3∞​2n−37n​

Use the binomial series to expand the function as a power series. Find the radius of convergence. 1(4+x)5\frac { 1 } { ( 4 + x ) ^ { 5 } }(4+x)51​

Find the value of the limit of the sequence defined by a1=1,an+1=6−1ana _ { 1 } = 1 , a _ { n + 1 } = 6 - \frac { 1 } { a _ { n } }a1​=1,an+1​=6−an​1​ .

Determine whether the given series converges or diverges. If it converges, find its sum. ∑n=1∞8n+78n+1\sum _ { n = 1 } ^ { \infty } \frac { 8 ^ { n } + 7 } { 8 ^ { n + 1 } }∑n=1∞​8n+18n+7​

Find the partial sum S7S _ { 7 }S7​ of the series ∑n=1∞13+4n\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 3 + 4 ^ { n } }∑n=1∞​3+4n1​ . Give your answer to five decimal places.

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } + 3 }$

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\sqrt [ 4 ] { 1 + x ^ { 6 } }$

Determine whether the given series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n } + 3 ^ { n } } { 9 ^ { n } }$

Test the series for convergence or divergence. $\sum _ { n = 1 } ^ { \infty } \sin \left( \frac { 1 } { n ^ { 4 } } \right)$

Find the value of the limit for the sequence. $\left\{ \arctan \left( \frac { 6 n } { 6 n + 8 } \right) \right\}$

Evaluate the indefinite integral as an infinite series. $\int \frac { 5 \left( e ^ { x } - 1 \right) } { x } d x$

Find an approximation of the sum of the series accurate to two decimal places. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ^ { 4 } }$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 2 } - 6 n + 14 }$

Use series to evaluate the limit correct to three decimal places. $\lim _ { x \rightarrow 0 } \frac { 7 x - \tan ^ { - 1 } 7 x } { x ^ { 3 } }$ Select the correct answer.

Determine whether the series converges or diverges. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } n } { 2 ^ { n } }$

Evaluate the function $f ( x ) = \cos x$ by a Taylor polynomial of degree $4$ centered at $a = 0$ , and $x = \frac { \pi } { 4 }$ .

Determine whether the sequence defined as follows is convergent or divergent. $a _ { 1 } = 1 , a _ { n + 1 } = 6 - a _ { n } \text { for } n \geq 1$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 6 } } { ( 5 n ) ! }$

Use the Comparison Test to determine whether the series is convergent or divergent. $\sum _ { n = 3 } ^ { \infty } \frac { 7 ^ { n } } { 2 ^ { n } - 3 }$

Use the binomial series to expand the function as a power series. Find the radius of convergence. $\frac { 1 } { ( 4 + x ) ^ { 5 } }$

Find the value of the limit of the sequence defined by $a _ { 1 } = 1 , a _ { n + 1 } = 6 - \frac { 1 } { a _ { n } }$ .

Determine whether the given series converges or diverges. If it converges, find its sum. $\sum _ { n = 1 } ^ { \infty } \frac { 8 ^ { n } + 7 } { 8 ^ { n + 1 } }$

Find the partial sum $S _ { 7 }$ of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { 3 + 4 ^ { n } }$ . Give your answer to five decimal places.