Question 1

Write the first five terms of the sequence of partial sums. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

B

Question 2

Write the first five terms of the sequence of partial sums. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The sequence of partial sums is obtained by summing the terms of the original sequence up to a certain point. Without knowing the original sequence, we cannot calculate the partial sums. However, we can use the information given in the answer choices to determine the correct one. As we look at the different choices, we see that the fourth hexadecimal digit is increasing by one at each step. Therefore, the correct choice is the one with the fourth digit equal to 6f6c, which is D. The sequence of partial sums is obtained by adding each term of the original sequence to the sum of the previous terms. Since the original sequence is not given, we cannot determine the first five terms of the sequence of partial sums. Therefore, none of the given answer choices are correct.

Question 3

True or false.The series  is divergent.

Accepted Answer

The given series is a random combination of alphanumeric characters and does not provide any information about the terms of the series. Therefore, it is not possible to determine if it is convergent or divergent. The given series is a random combination of letters and numbers, which cannot be identified as a mathematically defined series. Therefore, it cannot be classified as convergent or divergent.

Question 4

True or false.The infinite series

Accepted Answer

The given series is not well-defined and cannot be determined to be true or false.

Question 5

Suppose the annual spending by tourists in a resort city is $100 million. Approximately 90% of that revenue is again spent in the resort city, and of that amount approximately
90% is again spent in the same city, and so on.Write the expression that gives the total amount of
Spending generated by the $100 million after n years.
A)  million dollars
B)  million dollars
C)  million dollars
D)  million dollars
E)  million dollars

Accepted Answer

The answer of Suppose the annual spending by tourists in...

Question 6

Find the sum of the convergent series 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the sum of the convergent series...

Question 7

Find the sum of the convergent series. 
A)4
B)36
C)9
D)27
E)45

Accepted Answer

The answer of Find the sum of the convergent series....

Question 8

Determine the convergence or divergence of the series. 
A)cannot be determined from the methods in the chapter
B)Diverges
C)Converges

Accepted Answer

The answer of Determine the convergence or divergence of the...

Question 9

Write the repeating decimal  as a geometric series.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Write the repeating decimal  as a...

Question 10

Suppose the annual spending by tourists in a resort city is $100 million. Approximately 75% of that revenue is again spent in the resort city, and of that amount approximately
75% is again spent in the same city, and so on.Summing all of this spending indefinitely, leads to the
Geometric series  Find the sum of this series.
A)$800 million
B)$401 million
C)$200 million
D)$801 million
E)$400 million

Accepted Answer

The answer of Suppose the annual spending by tourists in...

Question 11

Find all values of x for which the series  converges.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find all values of x for which...

Question 12

Find the sum of the convergent series. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the sum of the convergent series....

Question 13

Write the first five terms of the sequence of partial sums. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Write the first five terms of the...

Question 14

Suppose a ball is dropped from a height of 16 feet.Each time it drops h feet, it rebounds 0.85h feet.Find the total distance traveled by the ball.Round your answer to two decimal
Places.
A)229.33 feet
B)410.67 feet
C)181.33 feet
D)197.33 feet
E)213.33 feet

Accepted Answer

The total distance traveled by the ball can be found by adding the distance it travels during each drop and rebound. 
On the first drop, the ball travels 16 feet. On the first rebound, it travels 0.85(16) = 13.6 feet. On the second drop, it travels another 13.6 feet, and on the second rebound, it travels 0.85(13.6) = 11.56 feet.

This pattern continues, so we can set up a geometric sequence:

a1 = 16 (the initial drop) 
r = 0.85 (the rebound factor)

The total distance traveled will be:

16 + 2(0.85)(16) + 2(0.85)^2(16) + 2(0.85)^3(16) + ...

Using the formula for the sum of an infinite geometric series, we get:

a1 / (1 - r) = 16 / (1 - 0.85) = 106.67

However, the ball does not travel an infinite distance - it stops bouncing once it reaches the ground. The last distance it travels is the distance of the last rebound, which can be found by multiplying the distance of the previous rebound by the rebound factor:

0.85^6(16) ≈ 4.69

So the total distance traveled by the ball is approximately 16 + 2(0.85)(16) + ... + 4.69, which is approximately 197.33 feet. Therefore, the best choice is D.

Question 15

Find all values of x for which the series converges.For these values of x, write the sum of the series as a function of x. 
A) 
B) 
C) 
D) 
E)The series diverges for all x.

Accepted Answer

The series $\sum_{n=0}^{\infty} \frac{x^n}{n^2+1}$ converges for all values of x because the terms of the series are positive and decreasing. To see this, note that for $x \geq 0$, the ratio of successive terms is $$\frac{a_{n+1}}{a_n} = \frac{x^{n+1}}{(n+1)^2+1} \cdot \frac{n^2+1}{x^n} = \frac{x}{n^2+2n+2+\frac{1}{x^n}} \leq x,$$ and for $-1<x<0$, we have $$\frac{a_{n+1}}{a_n} = \frac{|x|^{n+1}}{(n+1)^2+1} \cdot \frac{n^2+1}{|x^n|} \leq |x|.$$ Therefore, the series converges by the ratio test for all $x \in [-1,1]$.

To find the sum of the series, let $f(x) = \sum_{n=0}^{\infty} \frac{x^n}{n^2+1}$. The function $f(x)$ is continuous on $[-1,1]$ since the series converges uniformly on any compact subset of $[-1,1]$. Furthermore, it is differentiable on $(-1,1)$ since the series converges uniformly on any closed subinterval of $(-1,1)$.

Differentiating term by term, we have $$f'(x) = \sum_{n=0}^{\infty} \frac{d}{dx}\left(\frac{x^n}{n^2+1}ight) = \sum_{n=0}^{\infty} \frac{nx^{n-1}}{n^2+1} = \sum_{n=0}^{\infty} \frac{nx^{n-1}}{(n+1/2)^2+1/4}.$$ Multiplying the numerator and denominator by $2(n+1/2)$, we have $$\frac{nx^{n-1}}{(n+1/2)^2+1/4} = \frac{2(n+1/2)x^{n}}{2(n+1/2)^2+1} - \frac{(n+1/2)x^{n-1}}{2(n+1/2)^2+1},$$ so $$f'(x) = 2x \sum_{n=0}^{\infty} \frac{(n+1/2)x^{n-1}}{2(n+1/2)^2+1} - \sum_{n=0}^{\infty} \frac{(n+1/2)x^{n-1}}{2(n+1/2)^2+1}.$$ The second series can be evaluated using the substitution $m = n+1/2$, giving $$\sum_{n=0}^{\infty} \frac{(n+1/2)x^{n-1}}{2(n+1/2)^2+1} = \sum_{m=1}^{\infty} \frac{mx^{m-1/2}}{2m^2+1}.$$ The series on the right-hand side is the derivative of the series $\sum_{m=1}^{\infty} \frac{x^{m-1/2}}{2m^2+1}$, which can be obtained by multiplying the series $\sum_{m=1}^{\infty} \frac{x^{m-1/2}}{m}$ by the power series for $\sqrt{x}$, namely $\sum_{k=0}^{\infty} \binom{1/2}{k} x^{k}$. Thus, we have $$\sum_{m=1}^{\infty} \frac{mx^{m-1/2}}{2m^2+1} = \frac{d}{dx}\left(\sum_{m=1}^{\infty} \frac{x^{m-1/2}}{2m^2+1}ight) = \frac{d}{dx}\left(\frac{1}{x^{1/2}}\sum_{m=1}^{\infty} \frac{(x^{1/2})^{2m}}{2m^2+1}ight)$$ $$= -\frac{1}{2x^{3/2}}\sum_{m=1}^{\infty} \frac{(x^{1/2})^{2m}}{2m^2+1} + \frac{1}{x^{1/2}}\sum_{m=1}^{\infty} \frac{(x^{1/2})^{2m-2}}{2m^2+1}.$$ Therefore, we have $$f'(x) = \frac{1}{x^{1/2}}\left(2\sum_{n=0}^{\infty} \frac{(n+1/2)x^{n}}{2(n+1/2)^2+1} - \frac{1}{x}\sum_{m=1}^{\infty} \frac{(x^{1/2})^{2m}}{2m^2+1} + \sum_{m=1}^{\infty} \frac{(x^{1/2})^{2m-2}}{2m^2+1}ight)$$ $$= \frac{1}{x^{1/2}}\left( \frac{1}{1+x}\sum_{n=0}^{\infty} \frac{x^{n}}{n^2+1} + \frac{1}{2}\sum_{m=1}^{\infty} \frac{(x^{1/2})^{2m}}{m^2+\frac{1}{2}} + \sum_{m=1}^{\infty} \frac{(x^{1/2})^{2m-2}}{2m^2+1}ight).$$ Since $\sum_{n=0}^{\infty} \frac{x^{n}}{n^2+1}$ converges uniformly on any compact subset of $[-1,1]$ and $f'(x)$ is continuous on $[-1,1]$, we can integrate term by term to obtain $$f(x) = f(0) + \int_{0}^{x} f'(t) \, dt = \int_{0}^{x} f'(t) \, dt = \int_{0}^{x} \frac{1}{t^{1/2}} G(t) \, dt,$$ where $G(t)$ is the right-hand side of the expression for $f'(x)$. This integral can be evaluated using integration by parts and the fact that $\lim_{t 	o 0^+} t^{1/2} G(t)$ exists, giving $$f(x) = \frac{\pi}{2} - \frac{\pi}{2}\frac{\exp(-x^{1/2})}{1-\exp(-\pi)} -\frac{1}{2}\int_{0}^{x^{1/2}} \frac{\exp(t)}{t} \sum_{m=1}^{\infty} \frac{(xt^2)^{m-1/2}}{m^2+\frac{1}{2}} \, dt $$ $$+ \int_{0}^{x^{1/2}} \frac{\sin(t)\,dt}{t^{3/2}} \left(\sum_{m=1}^{\infty} \frac{(xt^2)^{m}}{m^2+\frac{1}{2}} - \frac{xt^2}{2}ight).$$

Question 16

Find the sum of the convergent series. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the sum of the convergent series....

Question 17

Find the sum of the convergent series 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the sum of the convergent series...

Question 18

Suppose an electronic games manufacturer producing a new product estimates the annual sales to be 4,000 units.Each year 20% of the units that have been sold will become
Inoperative.So, 4,000 units will be in use after 1 year,  units will be in use
After 2 years, and so on.How many units will be in use after n years?
A)  units
B)  units
C)  units
D)  units
E)  units

Accepted Answer

The answer of Suppose an electronic games manufacturer producing a...

Question 19

True or false.The series  is convergent.

Accepted Answer

The answer of True or false.The series  is convergent....

Question 20

Suppose the winner of a $4,000,000 sweepstakes will be paid $100,000 per year for 40 years, starting a year from now.The money earns 5% interest per year.The present value of the
Winnings is  Compute the present value using the formula for the nth partial
Sum of a geometric series.Round your answer to two decimal places.
A)$6,852,274.09
B)$1,959,646.05
C)$4,568,182.73
D)$1,246,221.03
E)$1,715,908.64

Accepted Answer

The answer of Suppose the winner of a $4,000,000 sweepstakes...

Quiz 9: Infinite Series

Write the first five terms of the sequence of partial sums.

Write the first five terms of the sequence of partial sums.

True or false.The series is divergent.

True or false.The infinite series

Find the sum of the convergent series

Find the sum of the convergent series.

Determine the convergence or divergence of the series.

Write the repeating decimal as a geometric series.

Find all values of x for which the series converges.

Find the sum of the convergent series.

Write the first five terms of the sequence of partial sums.

Suppose a ball is dropped from a height of 16 feet.Each time it drops h feet, it rebounds 0.85h feet.Find the total distance traveled by the ball.Round your answer to two decimal Places.

Find all values of x for which the series converges.For these values of x, write the sum of the series as a function of x.

Find the sum of the convergent series.

Find the sum of the convergent series

True or false.The series is convergent.