Question 1

Test the following series for convergence or divergence: $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { n ^ { 2 } } { n ^ { 2 } + 1 }$ .

Accepted Answer

Diverges b

Question 2

Find a power series representation for $\ln \left( 1 + x ^ { 2 } \right)$ .Hint: What is $\frac { d } { d x } \ln \left( 1 + x ^ { 2 } \right)$ ?

Accepted Answer

The answer of Find a power series representation for \(\ln...

Question 3

Which of the following series are convergent, but not absolutely convergent?
1) $\sum _ { n = 1 } ^ { \infty } ( - e ) ^ { - n }$ 
2) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { - n } n ^ { - 1 }$ 
3) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { - n } n ^ { - 2 }$

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 4

Find the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( 5 n - 1 ) ( 5 n + 1 ) }$ .

Accepted Answer

A)  $\frac { 1 } { 5 }$ 
B)  $\frac { 1 } { 6 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $\frac { 1 } { 10 }$ 
E) 1
F) $\frac { 3 } { 4 }$ 
G) $\frac { 3 } { 2 }$ 
H)Divergent 
A)  $\frac { 1 } { 5 }$ 
B)  $\frac { 1 } { 6 }$ 
C)  $\frac { 1 } { 2 }$ 
D)  $\frac { 1 } { 10 }$ 
E) 1
F) $\frac { 3 } { 4 }$ 
G) $\frac { 3 } { 2 }$ 
H)Divergent

Question 5

Given $f ( x ) = \tan x \text { and } a = 0$ ,
(a) calculate $T _ { 1 } ( x ; 0 )$ .(b) calculate $T _ { 3 } ( x ; 0 )$ .(c) calculate $T _ { 5 } ( x ; 0 )$ .

Accepted Answer

The answer of Given \(f ( x ) = \tan...

Question 6

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

Accepted Answer

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

Question 7

Suppose a 600 milligram dose of a drug is injected into a patient and that the patient's kidneys remove 20% of the drug from the bloodstream every hour. Let D(n) denote the amount of the drug left in the patient's body after n hours.(a) Find an expression for D(n).(b) How long will it take for the drug level to drop below 200 milligrams?
(c) How long will it take to bring the drug level below 10% of the original dosage?

Accepted Answer

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

Question 8

Estimate $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 4 } + 1 }$ to within 0.01.

Accepted Answer

The answer of Estimate \(\sum _ { n = 1...

Question 9

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

Accepted Answer

@#IMG-DLM& ; that is

Question 10

Use the binomial series to expand $\sqrt [ 5 ] { x - 1 }$ as a power series. State the radius of convergence.

Accepted Answer

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

Question 11

What is the smallest value of n that will guarantee (according to Taylor's Formula) that the Taylor polynomial $T _ { n }$ at the number 0 will be within 0.0001 of $e ^ { x }$ for $0 \leq x \leq 1$ ?

Accepted Answer

A) 4 
B) 5 
C) 8 
D) 6 
E) 7
F)2
G)3
H)9 
A) 4 
B) 5 
C) 8 
D) 6 
E) 7
F)2
G)3
H)9

Question 12

Test the following series for convergence or divergence: $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \sqrt { n + 3 } }$ .

Accepted Answer

Converges

Question 13

Here is a two-player game: Two players take turns flipping a fair coin. The first player to get a head wins the game. What is the probability that the person who starts first wins the game?

Accepted Answer

The answer of Here is a two-player game: Two players...

Question 14

Consider the recursive sequence defined by $a _ { 1 } = 1 ; a _ { n + 1 } = \frac { 1 } { 2 } \left( a _ { n } + 4 \right) , n > 1$ .(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) Use mathema

Question 15

Find a power series representation for the function $f ( x ) = \frac { x } { x + 5 }$ and find its interval of convergence.

Accepted Answer

@#IMG-DLM& , interva

Question 16

If $f ( x ) = e ^ { x ^ { 2 } }$ , compute $f ^ { 10 \rangle } ( 0 )$ .

Accepted Answer

The answer of If \(f ( x ) = e...

Question 17

Find the first four terms in the Maclaurin series for $f ( x ) = x e ^ { - x }$ .

Accepted Answer

A)  $x - x ^ { 2 } + x ^ { 3 } - x ^ { 4 }$ 
B)  $x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } - \frac { 1 } { 4 } x ^ { 4 }$ 
C)  $x - x ^ { 2 } + \frac { 1 } { 2 } x ^ { 3 } - \frac { 1 } { 6 } x ^ { 4 }$ 
D)  $x - 2 x ^ { 2 } + 3 x ^ { 3 } - 4 x ^ { 4 }$ 
E)  $x + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 6 } x ^ { 3 } + \frac { 1 } { 24 } x ^ { 4 }$ 
F) $x + x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } + \frac { 1 } { 8 } x ^ { 4 }$ 
G) $x + x ^ { 2 } + \frac { 1 } { 2 } x ^ { 3 } + \frac { 1 } { 6 } x ^ { 4 }$ 
H)$\frac { 1 } { 2 } x - \frac { 1 } { 6 } x ^ { 2 } + \frac { 1 } { 24 } x ^ { 3 } - \frac { 1 } { 120 } x ^ { 4 }$ 
A)  $x - x ^ { 2 } + x ^ { 3 } - x ^ { 4 }$ 
B)  $x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } - \frac { 1 } { 4 } x ^ { 4 }$ 
C)  $x - x ^ { 2 } + \frac { 1 } { 2 } x ^ { 3 } - \frac { 1 } { 6 } x ^ { 4 }$ 
D)  $x - 2 x ^ { 2 } + 3 x ^ { 3 } - 4 x ^ { 4 }$ 
E)  $x + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 6 } x ^ { 3 } + \frac { 1 } { 24 } x ^ { 4 }$ 
F) $x + x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } + \frac { 1 } { 8 } x ^ { 4 }$ 
G) $x + x ^ { 2 } + \frac { 1 } { 2 } x ^ { 3 } + \frac { 1 } { 6 } x ^ { 4 }$ 
H)$\frac { 1 } { 2 } x - \frac { 1 } { 6 } x ^ { 2 } + \frac { 1 } { 24 } x ^ { 3 } - \frac { 1 } { 120 } x ^ { 4 }$

Question 18

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

Accepted Answer

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

Question 19

Find the Taylor polynomial of degree 4 at 0 for the function defined by $f ( x ) = \ln ( 1 + x )$ . Then compute the value of $\ln ( 1.1 )$ accurate to as many decimal places as the polynomial of degree 4 allows.

Accepted Answer

@#IMG-DLM& , accurat

Question 20

Determine if the series $\sum _ { n = 1 } ^ { \infty } \left( \frac { 3 n - 1 } { 4 n + 1 } \right) ^ { n }$ converges or diverges by the Ratio Test or Root Test.

Accepted Answer

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

Test the following series for convergence or divergence: $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { n ^ { 2 } } { n ^ { 2 } + 1 }$ .

Find a power series representation for $\ln \left( 1 + x ^ { 2 } \right)$ .Hint: What is $\frac { d } { d x } \ln \left( 1 + x ^ { 2 } \right)$ ?

Which of the following series are convergent, but not absolutely convergent? 1) $\sum _ { n = 1 } ^ { \infty } ( - e ) ^ { - n }$ 2) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { - n } n ^ { - 1 }$ 3) $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { - n } n ^ { - 2 }$

Find the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( 5 n - 1 ) ( 5 n + 1 ) }$ .

Given $f ( x ) = \tan x \text { and } a = 0$ , (a) calculate $T _ { 1 } ( x ; 0 )$ .(b) calculate $T _ { 3 } ( x ; 0 )$ .(c) calculate $T _ { 5 } ( x ; 0 )$ .

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

Estimate $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 4 } + 1 }$ to within 0.01.

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

Use the binomial series to expand $\sqrt [ 5 ] { x - 1 }$ as a power series. State the radius of convergence.

What is the smallest value of n that will guarantee (according to Taylor's Formula) that the Taylor polynomial $T _ { n }$ at the number 0 will be within 0.0001 of $e ^ { x }$ for $0 \leq x \leq 1$ ?

Test the following series for convergence or divergence: $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \sqrt { n + 3 } }$ .

Here is a two-player game: Two players take turns flipping a fair coin. The first player to get a head wins the game. What is the probability that the person who starts first wins the game?

Consider the recursive sequence defined by $a _ { 1 } = 1 ; a _ { n + 1 } = \frac { 1 } { 2 } \left( a _ { n } + 4 \right) , n > 1$ .(a) Evaluate the first four terms of this sequence.(b) Show that the sequence converges.(c) Find the limit.

Find a power series representation for the function $f ( x ) = \frac { x } { x + 5 }$ and find its interval of convergence.

If $f ( x ) = e ^ { x ^ { 2 } }$ , compute $f ^ { 10 \rangle } ( 0 )$ .

Find the first four terms in the Maclaurin series for $f ( x ) = x e ^ { - x }$ .

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

Find the Taylor polynomial of degree 4 at 0 for the function defined by $f ( x ) = \ln ( 1 + x )$ . Then compute the value of $\ln ( 1.1 )$ accurate to as many decimal places as the polynomial of degree 4 allows.

Determine if the series $\sum _ { n = 1 } ^ { \infty } \left( \frac { 3 n - 1 } { 4 n + 1 } \right) ^ { n }$ converges or diverges by the Ratio Test or Root Test.

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

Exam 8: Infinite Sequences and Series

Test the following series for convergence or divergence: ∑n=1∞(−1)nn2n2+1\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { n ^ { 2 } } { n ^ { 2 } + 1 }∑n=1∞​(−1)nn2+1n2​ .

Find a power series representation for ln⁡(1+x2)\ln \left( 1 + x ^ { 2 } \right)ln(1+x2) .Hint: What is ddxln⁡(1+x2)\frac { d } { d x } \ln \left( 1 + x ^ { 2 } \right)dxd​ln(1+x2) ?

Find the sum of the series ∑n=1∞1(5n−1)(5n+1)\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( 5 n - 1 ) ( 5 n + 1 ) }∑n=1∞​(5n−1)(5n+1)1​ .

Given f(x)=tan⁡x and a=0f ( x ) = \tan x \text { and } a = 0f(x)=tanx and a=0 , (a) calculate T1(x;0)T _ { 1 } ( x ; 0 )T1​(x;0) .(b) calculate T3(x;0)T _ { 3 } ( x ; 0 )T3​(x;0) .(c) calculate T5(x;0)T _ { 5 } ( x ; 0 )T5​(x;0) .

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

Estimate ∑n=1∞1n4+1\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 4 } + 1 }∑n=1∞​n4+11​ to within 0.01.

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

Use the binomial series to expand x−15\sqrt [ 5 ] { x - 1 }5x−1​ as a power series. State the radius of convergence.

What is the smallest value of n that will guarantee (according to Taylor's Formula) that the Taylor polynomial TnT _ { n }Tn​ at the number 0 will be within 0.0001 of exe ^ { x }ex for 0≤x≤10 \leq x \leq 10≤x≤1 ?

Test the following series for convergence or divergence: ∑n=1∞(−1)nn+3\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \sqrt { n + 3 } }∑n=1∞​n+3​(−1)n​ .

Here is a two-player game: Two players take turns flipping a fair coin. The first player to get a head wins the game. What is the probability that the person who starts first wins the game?

Consider the recursive sequence defined by a1=1;an+1=12(an+4),n>1a _ { 1 } = 1 ; a _ { n + 1 } = \frac { 1 } { 2 } \left( a _ { n } + 4 \right) , n > 1a1​=1;an+1​=21​(an​+4),n>1 .(a) Evaluate the first four terms of this sequence.(b) Show that the sequence converges.(c) Find the limit.

Find a power series representation for the function f(x)=xx+5f ( x ) = \frac { x } { x + 5 }f(x)=x+5x​ and find its interval of convergence.

If f(x)=ex2f ( x ) = e ^ { x ^ { 2 } }f(x)=ex2 , compute f10⟩(0)f ^ { 10 \rangle } ( 0 )f10⟩(0) .

Find the first four terms in the Maclaurin series for f(x)=xe−xf ( x ) = x e ^ { - x }f(x)=xe−x .

Determine whether the series ∑n=1∞n3n+2\sum _ { n = 1 } ^ { \infty } \frac { n } { \sqrt { 3 n + 2 } }∑n=1∞​3n+2​n​ is convergent or divergent. If it is convergent, find the sum.

Find the Taylor polynomial of degree 4 at 0 for the function defined by f(x)=ln⁡(1+x)f ( x ) = \ln ( 1 + x )f(x)=ln(1+x) . Then compute the value of ln⁡(1.1)\ln ( 1.1 )ln(1.1) accurate to as many decimal places as the polynomial of degree 4 allows.

Determine if the series ∑n=1∞(3n−14n+1)n\sum _ { n = 1 } ^ { \infty } \left( \frac { 3 n - 1 } { 4 n + 1 } \right) ^ { n }∑n=1∞​(4n+13n−1​)n converges or diverges by the Ratio Test or Root Test.

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

Test the following series for convergence or divergence: $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } \frac { n ^ { 2 } } { n ^ { 2 } + 1 }$ .

Find a power series representation for $\ln \left( 1 + x ^ { 2 } \right)$ .Hint: What is $\frac { d } { d x } \ln \left( 1 + x ^ { 2 } \right)$ ?

Find the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( 5 n - 1 ) ( 5 n + 1 ) }$ .

Given $f ( x ) = \tan x \text { and } a = 0$ , (a) calculate $T _ { 1 } ( x ; 0 )$ .(b) calculate $T _ { 3 } ( x ; 0 )$ .(c) calculate $T _ { 5 } ( x ; 0 )$ .

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

Estimate $\sum _ { n = 1 } ^ { \infty } \frac { 1 } { n ^ { 4 } + 1 }$ to within 0.01.

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

Use the binomial series to expand $\sqrt [ 5 ] { x - 1 }$ as a power series. State the radius of convergence.

What is the smallest value of n that will guarantee (according to Taylor's Formula) that the Taylor polynomial $T _ { n }$ at the number 0 will be within 0.0001 of $e ^ { x }$ for $0 \leq x \leq 1$ ?

Test the following series for convergence or divergence: $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { \sqrt { n + 3 } }$ .

Consider the recursive sequence defined by $a _ { 1 } = 1 ; a _ { n + 1 } = \frac { 1 } { 2 } \left( a _ { n } + 4 \right) , n > 1$ .(a) Evaluate the first four terms of this sequence.(b) Show that the sequence converges.(c) Find the limit.

Find a power series representation for the function $f ( x ) = \frac { x } { x + 5 }$ and find its interval of convergence.

If $f ( x ) = e ^ { x ^ { 2 } }$ , compute $f ^ { 10 \rangle } ( 0 )$ .

Find the first four terms in the Maclaurin series for $f ( x ) = x e ^ { - x }$ .

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

Find the Taylor polynomial of degree 4 at 0 for the function defined by $f ( x ) = \ln ( 1 + x )$ . Then compute the value of $\ln ( 1.1 )$ accurate to as many decimal places as the polynomial of degree 4 allows.

Determine if the series $\sum _ { n = 1 } ^ { \infty } \left( \frac { 3 n - 1 } { 4 n + 1 } \right) ^ { n }$ converges or diverges by the Ratio Test or Root Test.