Question 1

Does $\sum_{n=1}^{\infty} \frac{1}{1+\sin n}$ converge?

Accepted Answer

no

Question 2

Does the series $\sum \frac{7 n}{n+8}$ converge or diverge.Explain.

Accepted Answer

It diverges because $\lim _{n \rightarrow \infty} \frac{7 n}{n+8}=7 \neq 0$ .

Question 3

Suppose the government spends $3.5 million on highways.Some of this money is earned by the highway workers who in turn spend $1,750,000 on food, travel, and entertainment.This causes $875,000 to be spent by the people who work in the food, travel, and entertainment industries.This $875,000 causes another $437,500 to be spent; the $437,500 causes another $218,750 to be spent, and so on.(Notice that each expenditure is half the previous one.)Assuming that this process continues forever, how many million dollars in total spending is generated by the original $3.5 million expenditure (including the original $3.5 million)?

Accepted Answer

7

Question 4

Stock prices for Abercrombie and Fitch fell steadily by an average of $0.94 per day from a high of $83.67 per share on December 24, 2007 to $70.05 on January 15, 2008.Let $P_{1}$ be the price of a share of stock on December 24, 2007.Write a formula for $P_{n}$ , the price of a share on the $n^{\text {th }}$ day after December 24.

Accepted Answer

The answer of Stock prices for Abercrombie and Fitch fell...

Question 5

If $\sum_{n=1}^{\infty} a_{n}$ converges then $\sum_{n=1}^{\infty} k a_{n}$ converges (k $\neq$ 0).

Accepted Answer

A) True 
 B)False

Question 6

Find the radius of convergence of $\sum_{n=0}^{\infty} \frac{6^{n+1}(x-1)^{n}}{\sqrt{n+1}}$ .

Accepted Answer

The radius

Question 7

Find the 6th partial sum of the series $\sum_{i=0}^{\infty}\left(\frac{5}{3}\right)^{i}$ (to two decimal places).

Accepted Answer

A) 26.12 
B)  $\infty$ 
C) 83.28 
D) 248.66 
A) 26.12 
B)  $\infty$ 
C) 83.28 
D) 248.66

Question 8

A radioactive isotope is released into the air as an industrial by-product.This isotope is not very stable due to radioactive decay.Two-thirds of the original radioactive material loses its radioactivity after each month.If 15 grams of this isotope are released into the atmosphere at the end of the first and every subsequent month, how many grams of radioactive material are in the atmosphere at the end of the twelfth month? Round to 2 decimal places.

Accepted Answer

The answer of A radioactive isotope is released into the...

Question 9

Does $\sum_{n=1}^{\infty} \frac{n^{4} \sin n}{n^{5}+4}$ converge?

Accepted Answer

The answer of Does $\sum_{n=1}^{\infty} \frac{n^{4} \sin n}{n^{5}+4}$ converge?...

Question 10

Use the integral test, if applicable, to determine whether the series $\sum_{n=1}^{\infty} \frac{n+2}{n^{2}+n}$ converges or diverges.

Accepted Answer

A) It converges. 
B) It diverges. 
C) The integral test does not apply. 
A) It converges. 
B) It diverges. 
C) The integral test does not apply.

Question 11

If a power series $\sum a_{k} x^{k}$ converges at x = c then it also converges at x = -c.

Accepted Answer

A) True 
 B)False

Question 12

Jamie was born in May.In August, her grandparents started a "Go to College in France" fund with $2200, earning a fixed annual interest rate of 7%.They added an additional $2200 each year in August until the last deposit in the year Jamie turned 18.Jamie estimated that she needed $90,000 to go start college in France.How much did she have in her "Go to College in France" fund? Did she have enough?

Accepted Answer

A)  $\sum_{n=0}^{18} 2200(1.07)^{n}=\$ 82233.72$ , no 
B)  $\sum_{n=0}^{18} 2200(1.07)^{n}=\$ 74797.87$ , no 
C) $91533.97, yes 
D) $87533.83, no 
E)  $\sum_{n=0}^{18} 2200(1.07)^{n+2}=\$ 74797.87$ , no 
A)  $\sum_{n=0}^{18} 2200(1.07)^{n}=\$ 82233.72$ , no 
B)  $\sum_{n=0}^{18} 2200(1.07)^{n}=\$ 74797.87$ , no 
C) $91533.97, yes 
D) $87533.83, no 
E)  $\sum_{n=0}^{18} 2200(1.07)^{n+2}=\$ 74797.87$ , no

Question 13

A tennis ball is dropped from a height of 15 feet and bounces.Each bounce is $\frac{1}{2}$ the height of the bounce before.A superball has a bounce $\frac{3}{4}$ the height of the bounce before, and is dropped from a height of 5 feet.Which ball bounces a greater total vertical distance?

Accepted Answer

A) The tennis ball 
B) The superball 
A) The tennis ball 
B) The superball

Question 14

Find the value of the infinite product $e^{1 / 8} \cdot e^{1 / 16} \cdot e^{1 / 32} \cdot e^{1 / 64} \cdot \ldots \cdot e^{1 / 2^{n+2}}$ to 2 decimal places.

Accepted Answer

The answer of Find the value of the infinite product...

Question 15

Find the radius of convergence of $x+\frac{3 x^{2}}{12}+\frac{4 x^{3}}{18}+\frac{5 x^{4}}{24}+\frac{6 x^{5}}{30}+\cdots$

Accepted Answer

The radius

Question 16

Find the sum of the series $\sum_{n=5}^{15}\left(\frac{4}{3}\right)^{n}$ to 2 decimal places.

Accepted Answer

The answer of Find the sum of the series $\sum_{n=5}^{15}\left(\frac{4}{3}\right)^{n}$...

Question 17

If a series of constants $\sum a_{k}$ diverges, then must $\sum\left|\alpha_{k}\right|$ diverge?

Accepted Answer

The answer of If a series of constants $\sum a_{k}$...

Question 18

Find an expression for the general term of the series $\frac{x}{5}+\frac{x^{2}}{12}+\frac{x^{3}}{31}+\frac{x^{4}}{68}+\cdots$

Accepted Answer

A)  $\frac{x^{n}}{n+4}$ for $n \geq 1$ . 
B)  $\frac{x^{n}}{7 n-2}$ for $n \geq 1$ . 
C)  $\frac{x^{n}}{n^{3}+4}$ for $n \geq 1$ . 
D)  $\frac{x^{n}}{n^{2}+12}$ for $n \geq 1$ . 
A)  $\frac{x^{n}}{n+4}$ for $n \geq 1$ . 
B)  $\frac{x^{n}}{7 n-2}$ for $n \geq 1$ . 
C)  $\frac{x^{n}}{n^{3}+4}$ for $n \geq 1$ . 
D)  $\frac{x^{n}}{n^{2}+12}$ for $n \geq 1$ .

Question 19

Use the integral test to decide whether the series $\sum_{n=1}^{\infty} \frac{5}{1+n^{2}}$ converges or diverges.

Accepted Answer

The answer of Use the integral test to decide whether...

Question 20

A radioactive isotope is released into the air as an industrial by-product.This isotope is not very stable due to radioactive decay.Two-thirds of the original radioactive material loses its radioactivity after each month.If 13 grams of this isotope are released into the atmosphere at the end of the first and every subsequent month and the situation goes on ad infinitum, how many grams of radioactive material are in the atmosphere at the end of each month in the long run?

Accepted Answer

The answer of A radioactive isotope is released into the...

Does $\sum_{n=1}^{\infty} \frac{1}{1+\sin n}$ converge?

Does the series $\sum \frac{7 n}{n+8}$ converge or diverge.Explain.

If $\sum_{n=1}^{\infty} a_{n}$ converges then $\sum_{n=1}^{\infty} k a_{n}$ converges (k $\neq$ 0).

Find the radius of convergence of $\sum_{n=0}^{\infty} \frac{6^{n+1}(x-1)^{n}}{\sqrt{n+1}}$ .

Find the 6th partial sum of the series $\sum_{i=0}^{\infty}\left(\frac{5}{3}\right)^{i}$ (to two decimal places).

Does $\sum_{n=1}^{\infty} \frac{n^{4} \sin n}{n^{5}+4}$ converge?

Use the integral test, if applicable, to determine whether the series $\sum_{n=1}^{\infty} \frac{n+2}{n^{2}+n}$ converges or diverges.

If a power series $\sum a_{k} x^{k}$ converges at x = c then it also converges at x = -c.

A tennis ball is dropped from a height of 15 feet and bounces.Each bounce is $\frac{1}{2}$ the height of the bounce before.A superball has a bounce $\frac{3}{4}$ the height of the bounce before, and is dropped from a height of 5 feet.Which ball bounces a greater total vertical distance?

Find the value of the infinite product $e^{1 / 8} \cdot e^{1 / 16} \cdot e^{1 / 32} \cdot e^{1 / 64} \cdot \ldots \cdot e^{1 / 2^{n+2}}$ to 2 decimal places.

Find the radius of convergence of $x+\frac{3 x^{2}}{12}+\frac{4 x^{3}}{18}+\frac{5 x^{4}}{24}+\frac{6 x^{5}}{30}+\cdots$

Find the sum of the series $\sum_{n=5}^{15}\left(\frac{4}{3}\right)^{n}$ to 2 decimal places.

If a series of constants $\sum a_{k}$ diverges, then must $\sum\left|\alpha_{k}\right|$ diverge?

Find an expression for the general term of the series $\frac{x}{5}+\frac{x^{2}}{12}+\frac{x^{3}}{31}+\frac{x^{4}}{68}+\cdots$

Use the integral test to decide whether the series $\sum_{n=1}^{\infty} \frac{5}{1+n^{2}}$ converges or diverges.

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Exam 9: Sequences and Series

Does ∑n=1∞11+sin⁡n\sum_{n=1}^{\infty} \frac{1}{1+\sin n}∑n=1∞​1+sinn1​ converge?

Does the series ∑7nn+8\sum \frac{7 n}{n+8}∑n+87n​ converge or diverge.Explain.

If ∑n=1∞an\sum_{n=1}^{\infty} a_{n}∑n=1∞​an​ converges then ∑n=1∞kan\sum_{n=1}^{\infty} k a_{n}∑n=1∞​kan​ converges (k ≠\neq= 0).

Find the radius of convergence of ∑n=0∞6n+1(x−1)nn+1\sum_{n=0}^{\infty} \frac{6^{n+1}(x-1)^{n}}{\sqrt{n+1}}∑n=0∞​n+1​6n+1(x−1)n​ .

Find the 6th partial sum of the series ∑i=0∞(53)i\sum_{i=0}^{\infty}\left(\frac{5}{3}\right)^{i}∑i=0∞​(35​)i (to two decimal places).

Does ∑n=1∞n4sin⁡nn5+4\sum_{n=1}^{\infty} \frac{n^{4} \sin n}{n^{5}+4}∑n=1∞​n5+4n4sinn​ converge?

Use the integral test, if applicable, to determine whether the series ∑n=1∞n+2n2+n\sum_{n=1}^{\infty} \frac{n+2}{n^{2}+n}∑n=1∞​n2+nn+2​ converges or diverges.

If a power series ∑akxk\sum a_{k} x^{k}∑ak​xk converges at x = c then it also converges at x = -c.

A tennis ball is dropped from a height of 15 feet and bounces.Each bounce is 12\frac{1}{2}21​ the height of the bounce before.A superball has a bounce 34\frac{3}{4}43​ the height of the bounce before, and is dropped from a height of 5 feet.Which ball bounces a greater total vertical distance?

Find the value of the infinite product e1/8⋅e1/16⋅e1/32⋅e1/64⋅…⋅e1/2n+2e^{1 / 8} \cdot e^{1 / 16} \cdot e^{1 / 32} \cdot e^{1 / 64} \cdot \ldots \cdot e^{1 / 2^{n+2}}e1/8⋅e1/16⋅e1/32⋅e1/64⋅…⋅e1/2n+2 to 2 decimal places.

Find the radius of convergence of x+3x212+4x318+5x424+6x530+⋯x+\frac{3 x^{2}}{12}+\frac{4 x^{3}}{18}+\frac{5 x^{4}}{24}+\frac{6 x^{5}}{30}+\cdotsx+123x2​+184x3​+245x4​+306x5​+⋯

Find the sum of the series ∑n=515(43)n\sum_{n=5}^{15}\left(\frac{4}{3}\right)^{n}∑n=515​(34​)n to 2 decimal places.

If a series of constants ∑ak\sum a_{k}∑ak​ diverges, then must ∑∣αk∣\sum\left|\alpha_{k}\right|∑∣αk​∣ diverge?

Find an expression for the general term of the series x5+x212+x331+x468+⋯\frac{x}{5}+\frac{x^{2}}{12}+\frac{x^{3}}{31}+\frac{x^{4}}{68}+\cdots5x​+12x2​+31x3​+68x4​+⋯

Use the integral test to decide whether the series ∑n=1∞51+n2\sum_{n=1}^{\infty} \frac{5}{1+n^{2}}∑n=1∞​1+n25​ converges or diverges.

A Library of Functions

Key Concept: the Derivative

Short-Cuts to Differentiation

Using the Derivative

Key Concept- the Definite Integral

Constructing Antiderivatives

Integration

Using the Definite Integral

Approximating Functions Using Series

Differential Equations

Functions of Several Variables

A Fundamental Tool- Vectors

Differentiating Functions of Several Variables

Optimization- Local and Global Extrema

Integrating Functions of Several Variables

Parameterization and Vector Fields

Line Integrals

Flux Integrals and Divergence

The Curl and Stokes Theorem

Parameters, Coordinates, Integrals

Filters

Does $\sum_{n=1}^{\infty} \frac{1}{1+\sin n}$ converge?

Does the series $\sum \frac{7 n}{n+8}$ converge or diverge.Explain.

If $\sum_{n=1}^{\infty} a_{n}$ converges then $\sum_{n=1}^{\infty} k a_{n}$ converges (k $\neq$ 0).

Find the radius of convergence of $\sum_{n=0}^{\infty} \frac{6^{n+1}(x-1)^{n}}{\sqrt{n+1}}$ .

Find the 6th partial sum of the series $\sum_{i=0}^{\infty}\left(\frac{5}{3}\right)^{i}$ (to two decimal places).

Does $\sum_{n=1}^{\infty} \frac{n^{4} \sin n}{n^{5}+4}$ converge?

Use the integral test, if applicable, to determine whether the series $\sum_{n=1}^{\infty} \frac{n+2}{n^{2}+n}$ converges or diverges.

If a power series $\sum a_{k} x^{k}$ converges at x = c then it also converges at x = -c.

A tennis ball is dropped from a height of 15 feet and bounces.Each bounce is $\frac{1}{2}$ the height of the bounce before.A superball has a bounce $\frac{3}{4}$ the height of the bounce before, and is dropped from a height of 5 feet.Which ball bounces a greater total vertical distance?

Find the value of the infinite product $e^{1 / 8} \cdot e^{1 / 16} \cdot e^{1 / 32} \cdot e^{1 / 64} \cdot \ldots \cdot e^{1 / 2^{n+2}}$ to 2 decimal places.

Find the radius of convergence of $x+\frac{3 x^{2}}{12}+\frac{4 x^{3}}{18}+\frac{5 x^{4}}{24}+\frac{6 x^{5}}{30}+\cdots$

Find the sum of the series $\sum_{n=5}^{15}\left(\frac{4}{3}\right)^{n}$ to 2 decimal places.

If a series of constants $\sum a_{k}$ diverges, then must $\sum\left|\alpha_{k}\right|$ diverge?

Find an expression for the general term of the series $\frac{x}{5}+\frac{x^{2}}{12}+\frac{x^{3}}{31}+\frac{x^{4}}{68}+\cdots$

Use the integral test to decide whether the series $\sum_{n=1}^{\infty} \frac{5}{1+n^{2}}$ converges or diverges.