Question 1

Find the Maclaurin series for f (x) using the definition of the Maclaurin series. $f ( x ) = x \cos ( 4 x )$

Accepted Answer

A)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 5 ^ { 2 n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$ 
B)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 5 ^ { 2 n } x ^ { 2 n + 1 } } { n ! }$ 
C)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } 5 ^ { 2 n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$ 
D)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 5 ^ { n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$ 
E)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 5 ^ { 2 n } x ^ { 2 n } } { ( 2 n ) ! }$ 
A)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 5 ^ { 2 n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$ 
B)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 5 ^ { 2 n } x ^ { 2 n + 1 } } { n ! }$ 
C)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n + 1 } 5 ^ { 2 n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$ 
D)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 5 ^ { n } x ^ { 2 n + 1 } } { ( 2 n ) ! }$ 
E)  $\sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } 5 ^ { 2 n } x ^ { 2 n } } { ( 2 n ) ! }$

Question 2

Find all positive values of u for which the series $\sum _ { m = 1 } ^ { \infty } 3 u ^ { \ln 9 m }$ converges.

Accepted Answer

A) $4 \ln 7$ D) $u > 7$ E) $u < 4$ A) $4 \ln 7$ D) $u > 7$ E) $u < 4$

Question 3

How many terms of the series do we need to add in order to find the sum to the indicated accuracy? $\sum _ { n = 1 } ^ { \infty } 2 \frac { ( - 1 ) ^ { n + 1 } } { n ^ { 2 } } ( \text { |error } \mid ) < 0.0399$

Accepted Answer

A)  $n = 6$ 
B)  $n = 8$ 
C)  $n = 5$ 
D)  $n = 12$ 
E)  $n = 13$ 
A)  $n = 6$ 
B)  $n = 8$ 
C)  $n = 5$ 
D)  $n = 12$ 
E)  $n = 13$

Question 4

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln \left( n ^ { 4 } \right) } { n } \right) ^ { n }$

Accepted Answer

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

Question 5

Use the Comparison Test to determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 5 + \sin 2 n } { 4 ^ { n } }$

Accepted Answer

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

Question 6

Use the Integral Test to determine whether the series is convergent or divergent. $\frac { 3 } { 17 } + \frac { 3 } { 26 } + \frac { 3 } { 37 } + \frac { 3 } { 50 } + \frac { 3 } { 65 } + \cdots$

Accepted Answer

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

Question 7

Test the series for convergence or divergence. $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { \sqrt { n ^ { 5 } + 8 } }$

Accepted Answer

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

Question 8

Use the Integral Test to determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 8 } { 5 n + 4 }$

Accepted Answer

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

Question 9

Find an approximation of the sum of the series accurate to two decimal places. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ^ { 3 } }$

Accepted Answer

A)  -0.83 
B)  -0.90 
C)  -0.96 
D)  -1.02 
A)  -0.83 
B)  -0.90 
C)  -0.96 
D)  -1.02

Question 10

Test the series for convergence or divergence. $\sum _ { n = 1 } ^ { \infty } \frac { 6 + \sin 5 n } { 7 ^ { n } }$

Accepted Answer

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

Question 11

Determine whether the sequence convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 5 } { n ^ { 2 } + 5 }$

Accepted Answer

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

Question 12

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 5 n } { 4 n + 1 } \right) ^ { n }$

Accepted Answer

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

Question 13

Determine whether the given series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } \frac { - 2 } { ( n + 3 ) ( n + 1 ) }$

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

An annuity is a sequence of payments made at regular intervals. Suppose that a sum of $200 is deposited at the end of each month into an account earning interest at the rate of 12% per year compounded monthly. Then the amount on deposit at the end of the $n ^ { \text {th } }$ month is $f ( n ) = 20,000 \left[ ( 1.01 ) ^ { n } - 1 \right]$ .
Consider the sequence $\left\{ a _ { n } \right\}$ defined by $a _ { n } = f ( n )$ (a) Find the $24 ^ { \text {th } }$ term of the sequence, and interpret your result.
(b) Evaluate $\lim _ { n \rightarrow \infty } a _ { n } ,$ and interpret your result.

Accepted Answer

(a) $5,394.69 -- This is the a

Question 16

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 5 x + 3 ) ^ { n } } { n \ln n }$

Accepted Answer

The answer of Find the radius of convergence and the...

Question 17

Find all values of p for which the series $\sum _ { n = 1 } ^ { \infty } \frac { \ln \left( n ^ { 9 } \right) } { n ^ { y } }$ converges.

Accepted Answer

A)  $p  9$ 
D)  $p > 1$ 
A)  $p  9$ 
D)  $p > 1$

Question 18

Which of the partial sums of the alternating series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { n }$ are overestimates of the total sum?

Accepted Answer

A)  $S _ { 88 }$ 
B)  $S _ { 98 }$ 
C)  $s _ { 65 }$ 
D)  $S _ { 77 }$ 
E)  $S _ { 89 }$ 
A)  $S _ { 88 }$ 
B)  $S _ { 98 }$ 
C)  $s _ { 65 }$ 
D)  $S _ { 77 }$ 
E)  $S _ { 89 }$

Question 19

If $600 is invested at $4 \%$ interest, compounded annually, then after n years the investment is worth $a _ { n } = 600 ( 1.04 ) ^ { n }$ dollars. Find the size of investment after 7 years.

Accepted Answer

A)  $430.21 
B)  $1,860.81 
C)  $1,230.81 
D)  $789.56 
E)  $1,321.06 
A)  $430.21 
B)  $1,860.81 
C)  $1,230.81 
D)  $789.56 
E)  $1,321.06

Question 20

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 4 n } { 3 n + 1 } \right) ^ { n }$

Accepted Answer

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

Find the Maclaurin series for f (x) using the definition of the Maclaurin series. $f ( x ) = x \cos ( 4 x )$

Find all positive values of u for which the series $\sum _ { m = 1 } ^ { \infty } 3 u ^ { \ln 9 m }$ converges.

How many terms of the series do we need to add in order to find the sum to the indicated accuracy? $\sum _ { n = 1 } ^ { \infty } 2 \frac { ( - 1 ) ^ { n + 1 } } { n ^ { 2 } } ( \text { |error } \mid ) < 0.0399$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln \left( n ^ { 4 } \right) } { n } \right) ^ { n }$

Use the Comparison Test to determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 5 + \sin 2 n } { 4 ^ { n } }$

Use the Integral Test to determine whether the series is convergent or divergent. $\frac { 3 } { 17 } + \frac { 3 } { 26 } + \frac { 3 } { 37 } + \frac { 3 } { 50 } + \frac { 3 } { 65 } + \cdots$

Test the series for convergence or divergence. $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { \sqrt { n ^ { 5 } + 8 } }$

Use the Integral Test to determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 8 } { 5 n + 4 }$

Find an approximation of the sum of the series accurate to two decimal places. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ^ { 3 } }$

Test the series for convergence or divergence. $\sum _ { n = 1 } ^ { \infty } \frac { 6 + \sin 5 n } { 7 ^ { n } }$

Determine whether the sequence convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 5 } { n ^ { 2 } + 5 }$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 5 n } { 4 n + 1 } \right) ^ { n }$

Determine whether the given series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } \frac { - 2 } { ( n + 3 ) ( n + 1 ) }$

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

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 5 x + 3 ) ^ { n } } { n \ln n }$

Find all values of p for which the series $\sum _ { n = 1 } ^ { \infty } \frac { \ln \left( n ^ { 9 } \right) } { n ^ { y } }$ converges.

Which of the partial sums of the alternating series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { n }$ are overestimates of the total sum?

If $600 is invested at $4 \%$ interest, compounded annually, then after n years the investment is worth $a _ { n } = 600 ( 1.04 ) ^ { n }$ dollars. Find the size of investment after 7 years.

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 4 n } { 3 n + 1 } \right) ^ { n }$

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

Find the Maclaurin series for f (x) using the definition of the Maclaurin series. f(x)=xcos⁡(4x)f ( x ) = x \cos ( 4 x )f(x)=xcos(4x)

Find all positive values of u for which the series ∑m=1∞3uln⁡9m\sum _ { m = 1 } ^ { \infty } 3 u ^ { \ln 9 m }∑m=1∞​3uln9m converges.

How many terms of the series do we need to add in order to find the sum to the indicated accuracy? ∑n=1∞2(−1)n+1n2( |error ∣)<0.0399\sum _ { n = 1 } ^ { \infty } 2 \frac { ( - 1 ) ^ { n + 1 } } { n ^ { 2 } } ( \text { |error } \mid ) < 0.0399∑n=1∞​2n2(−1)n+1​( |error ∣)<0.0399

Determine whether the series is convergent or divergent. ∑n=1∞(ln⁡(n4)n)n\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln \left( n ^ { 4 } \right) } { n } \right) ^ { n }∑n=1∞​(nln(n4)​)n

Use the Comparison Test to determine whether the series is convergent or divergent. ∑n=1∞5+sin⁡2n4n\sum _ { n = 1 } ^ { \infty } \frac { 5 + \sin 2 n } { 4 ^ { n } }∑n=1∞​4n5+sin2n​

Use the Integral Test to determine whether the series is convergent or divergent. 317+326+337+350+365+⋯\frac { 3 } { 17 } + \frac { 3 } { 26 } + \frac { 3 } { 37 } + \frac { 3 } { 50 } + \frac { 3 } { 65 } + \cdots173​+263​+373​+503​+653​+⋯

Test the series for convergence or divergence. ∑n=0∞1n5+8\sum _ { n = 0 } ^ { \infty } \frac { 1 } { \sqrt { n ^ { 5 } + 8 } }∑n=0∞​n5+8​1​

Use the Integral Test to determine whether the series is convergent or divergent. ∑n=1∞85n+4\sum _ { n = 1 } ^ { \infty } \frac { 8 } { 5 n + 4 }∑n=1∞​5n+48​

Find an approximation of the sum of the series accurate to two decimal places. ∑n=1∞(−1)nn3\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ^ { 3 } }∑n=1∞​n3(−1)n​

Test the series for convergence or divergence. ∑n=1∞6+sin⁡5n7n\sum _ { n = 1 } ^ { \infty } \frac { 6 + \sin 5 n } { 7 ^ { n } }∑n=1∞​7n6+sin5n​

Determine whether the sequence convergent or divergent. ∑n=1∞5n2+5\sum _ { n = 1 } ^ { \infty } \frac { 5 } { n ^ { 2 } + 5 }∑n=1∞​n2+55​

Determine whether the series is convergent or divergent. ∑n=1∞(5n4n+1)n\sum _ { n = 1 } ^ { \infty } \left( \frac { 5 n } { 4 n + 1 } \right) ^ { n }∑n=1∞​(4n+15n​)n

Determine whether the given series converges or diverges. If it converges, find its sum. ∑n=0∞−2(n+3)(n+1)\sum _ { n = 0 } ^ { \infty } \frac { - 2 } { ( n + 3 ) ( n + 1 ) }∑n=0∞​(n+3)(n+1)−2​

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

Find the radius of convergence and the interval of convergence of the power series. ∑n=2∞(−1)n(5x+3)nnln⁡n\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 5 x + 3 ) ^ { n } } { n \ln n }∑n=2∞​nlnn(−1)n(5x+3)n​

Find all values of p for which the series ∑n=1∞ln⁡(n9)ny\sum _ { n = 1 } ^ { \infty } \frac { \ln \left( n ^ { 9 } \right) } { n ^ { y } }∑n=1∞​nyln(n9)​ converges.

Which of the partial sums of the alternating series ∑n=1∞(−1)n−1n\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { n }∑n=1∞​n(−1)n−1​ are overestimates of the total sum?

If $600 is invested at 4%4 \%4% interest, compounded annually, then after n years the investment is worth an=600(1.04)na _ { n } = 600 ( 1.04 ) ^ { n }an​=600(1.04)n dollars. Find the size of investment after 7 years.

Determine whether the series is convergent or divergent. ∑n=1∞(4n3n+1)n\sum _ { n = 1 } ^ { \infty } \left( \frac { 4 n } { 3 n + 1 } \right) ^ { n }∑n=1∞​(3n+14n​)n

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

Find the Maclaurin series for f (x) using the definition of the Maclaurin series. $f ( x ) = x \cos ( 4 x )$

Find all positive values of u for which the series $\sum _ { m = 1 } ^ { \infty } 3 u ^ { \ln 9 m }$ converges.

How many terms of the series do we need to add in order to find the sum to the indicated accuracy? $\sum _ { n = 1 } ^ { \infty } 2 \frac { ( - 1 ) ^ { n + 1 } } { n ^ { 2 } } ( \text { |error } \mid ) < 0.0399$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { \ln \left( n ^ { 4 } \right) } { n } \right) ^ { n }$

Use the Comparison Test to determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 5 + \sin 2 n } { 4 ^ { n } }$

Use the Integral Test to determine whether the series is convergent or divergent. $\frac { 3 } { 17 } + \frac { 3 } { 26 } + \frac { 3 } { 37 } + \frac { 3 } { 50 } + \frac { 3 } { 65 } + \cdots$

Test the series for convergence or divergence. $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { \sqrt { n ^ { 5 } + 8 } }$

Use the Integral Test to determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 8 } { 5 n + 4 }$

Find an approximation of the sum of the series accurate to two decimal places. $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { n ^ { 3 } }$

Test the series for convergence or divergence. $\sum _ { n = 1 } ^ { \infty } \frac { 6 + \sin 5 n } { 7 ^ { n } }$

Determine whether the sequence convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \frac { 5 } { n ^ { 2 } + 5 }$

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 5 n } { 4 n + 1 } \right) ^ { n }$

Determine whether the given series converges or diverges. If it converges, find its sum. $\sum _ { n = 0 } ^ { \infty } \frac { - 2 } { ( n + 3 ) ( n + 1 ) }$

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

Find the radius of convergence and the interval of convergence of the power series. $\sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 5 x + 3 ) ^ { n } } { n \ln n }$

Find all values of p for which the series $\sum _ { n = 1 } ^ { \infty } \frac { \ln \left( n ^ { 9 } \right) } { n ^ { y } }$ converges.

Which of the partial sums of the alternating series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { n }$ are overestimates of the total sum?

If $600 is invested at $4 \%$ interest, compounded annually, then after n years the investment is worth $a _ { n } = 600 ( 1.04 ) ^ { n }$ dollars. Find the size of investment after 7 years.

Determine whether the series is convergent or divergent. $\sum _ { n = 1 } ^ { \infty } \left( \frac { 4 n } { 3 n + 1 } \right) ^ { n }$