5. Use the comparison test or limit comparison test to determine whether the following series is convergent or divergent. 1 (a) Σn=2 1/vn-1 (b) Σn=1 4sin2n/n!
Answers
The series Σ(n=2 to ∞) 1/√(n-1) is also convergent by the comparison test and the limit comparison test, the series\(Σ(n=1 to ∞) (4sin^2(n))/n!\)is convergent.
(a) To determine the convergence of the series Σ(n=2 to ∞) 1/√(n-1) using the comparison test, we can compare it with a known series. A common choice is the p-series, Σ(n=1 to ∞) 1/n^p, where p>1 for convergence.
In this case, we compare our series to Σ(n=2 to ∞) 1/n^(1/2), which is a convergent p-series with p=1/2. Now, we examine the ratio:
\(1/√(n-1) / 1/n^(1/2) = √n / √(n-1)\)
As n approaches infinity, this ratio approaches 1, which means the series Σ(n=2 to ∞) 1/√(n-1) is also convergent by the comparison test.
(b) To determine the convergence of the series \(Σ(n=1 to ∞) (4sin^2(n))/n!\), we can use the limit comparison test. Since n! grows much faster than any power of n, we can compare our series with the simpler series Σ(n=1 to ∞) 1/n!.
Now, examine the limit:
\(lim (n → ∞) [(4sin^2(n))/n!] / [1/n!] = lim (n → ∞) 4sin^2(n)\)
Since the sine function is bounded between -1 and 1, the limit of 4sin^2(n) is also bounded between 0 and 4 as n approaches infinity. Therefore, by the limit comparison test, the series Σ(n=1 to ∞) (4sin^2(n))/n! is convergent.
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Related Questions
What is 0.3666... (the 6 is repeating) expressed as a fraction?
Here is what I have so far, i dont know if It is helpful:
If X = 0.36666…, then 100x = 36.6666… and 10x = 3.6666…. We can subtract 10x from 100x to get 90x, which is equal to 36.6666 – 3.6666 or 33. That means 90x = 33.
Answers
Answer:
33/90 is the answer
Step-by-step explanation:
exactly what you have there is the procedure
Three runners are training for a marathon. one day, they all run about ten miles, each at their own constant speed. which of the three runners has the fastest pace?
Answers
You did not post enough information.
Triangle ABC is translated according to the rule (x, y) → (x + 2, y – 8). If the coordinates of the pre-image of point B are
(4, –5), what are the coordinates of B'?
Answers
Answer:
B'(6,-3)
Step-by-step explanation:
Answer:
its B on edgenuity
Step-by-step explanation:
I need help now if baby yoda is to live in the next episode in Mandalorian
Answers
Answer:
1. 52*
2. 128* ( I think)
Step-by-step explanation:
in your own words, describe what the sampling distribution of the mean is. in your answer, be sure to address the shape of the distribution [1 point] and the sampling used to create the distribution [1 point]. additionally, address the mean [1 point] and variability of the distribution
Answers
The sampling distribution of the mean is approximately normal, follows the Central Limit Theorem, has a mean equal to the population mean, and decreases in variability with larger sample sizes.
The sampling distribution of the mean is a distribution that represents the possible values of the sample means obtained from repeated sampling of a population. It tends to follow a normal curve shape, regardless of the shape of the population distribution, as long as the sample size is large enough due to the Central Limit Theorem.
The mean of the sampling distribution is equal to the population mean, making it an unbiased estimator. Additionally, the standard deviation of the sampling distribution, known as the standard error, decreases as the sample size increases, indicating that larger samples lead to more precise estimates of the population mean with reduced variability in the sample means.
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yuson thinks the design she made is symmetric across the dashed line she drew. How can she use coordinates to show that her design is symmetric?
Answers
She can compare the original design with its reflection by plotting both designs on the same graph. If the two designs are identical, then the design is symmetric across the x-axis.
How to show if her design is symmetricTo show that a design is symmetric across a line using coordinates, you can use the following steps:
Choose the equation of the line that the design is supposed to be symmetric across. Let's assume that the line is the x-axis and it is represented by the equation y = 0.Identify the coordinates of each point in the design. For example, if a point is located at (2, 4), its x-coordinate is 2 and its y-coordinate is 4.Reflect each point in the design across the line of symmetry by changing the sign of its y-coordinate. For example, if a point is located at (2, 4), its reflection across the x-axis is located at (2, -4).Compare the original design with its reflection across the line of symmetry. If the reflected design is identical to the original design, then the design is symmetric across the line.Here's an example of how to use coordinates to show that a design is symmetric across the x-axis:
Suppose Yuson has a design with three points: A(2, 4), B(4, 6), and C(6, 4), and she drew a dashed line along the x-axis to represent the line of symmetry.
To check if the design is symmetric across the x-axis, she can reflect each point across the x-axis to obtain the following new coordinates: A'(2, -4), B'(4, -6), and C'(6, -4).
She can then compare the original design with its reflection by plotting both designs on the same graph. If the two designs are identical, then the design is symmetric across the x-axis.
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solve the initial-value problem. 2xy' y = 6x, x > 0, y(4) = 24
Answers
To solve the initial-value problem 2xy' y = 6x, x > 0, y(4) = 24, we first need to separate the variables and integrate both sides.
Starting with the initial-value problem 2xy' y = 6x, we divide both sides by 2xy to get y'/y = 3/x.
Now, we integrate both sides with respect to x:
∫(y'/y) dx = ∫(3/x) dx
=> ln|y| = 3ln|x| + C
where C is the constant of integration.
Next, we can solve for y by exponentiating both sides:
|y| = e^(3ln|x|+C)
=>|y| = e^C * e^(ln|x|^3)
=>|y| = kx^3
where k = ± e^C.
To find the specific value of k, we can use the initial condition y(4) = 24:
|24| = k(4)^3
=>k = 3/8
So, the solution to the initial-value problem is:
y = 3/8 x^3 for x > 0, y(4) = 24.
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Could you please check my answer? Use the equations to solve the system of equations:Y=0X=7My answer: (7,0)
Answers
ANSWER:
\((7,0)\)STEP-BY-STEP EXPLANATION:
We have the following system of equations
\(\begin{gathered} x=7 \\ y=0 \end{gathered}\)In the case x = 7, it is parallel to the y axis, through the point 7 in x, and in the case y = 0, it is the same x axis, therefore the intercept is the point (7 , 0)
The solution is:
\((7,0)\)PLEASE HELP ME!!!!!!!
Slowhan decides to play a very fun game. He starts off with his favorite
odd integer. For every turn, Slowhan will either add 1 or subtract 1 to his current number.
What is the probability that after 11 turns, the resulting number will be odd?
Answers
Answer:
Answer is even on the 11th turn unless Slowhan is allowed to skip a turn.
Step-by-step explanation:
If you add or subtract every turn then every turn will always be the opposite of the last turn, so, it would be.
odd is 0
1 = even
2 = odd (You've seen here that every even numbers of turns how answer will be odd and when it is an odd number of turns it will be even)
3 = even
4 = odd
5 = even
6 = odd
7 = even
8 = odd
9 = even
10 = odd
11 = even
Consider the curve described by the vector function r(t)=t cos ti+t sin tj+2tk. (a) Show that motion along this curve occurs at an increasing speed as t>0 increases. Hint: The speed at a point is given by the length of the tangent vector at that point.) (b) Find parametric equations for the line tangent to this curve at the point (0,π/2,π).
Answers
(a) To show that motion along the curve described by the vector function \(\( r(t) = t \cos(t)i + t \sin(t)j + 2tk \)\) occurs at an increasing speed as t > 0 increases, we need to find the speed function
(a) The speed at a point on the curve is given by the magnitude of the tangent vector at that point. The derivative of the position vector r(t) with respect to t gives the tangent vector r'(t). The speed function is given by r'(t) , the magnitude of r'(t). By finding the derivative of the speed function with respect to t and showing that it is positive for t > 0 , we can conclude that motion along the curve occurs at an increasing speed as t increases.
(b) To find the parametric equations for the line tangent to the curve at the point \(\((0, \frac{\pi}{2}, \pi)\)\), we need to find the derivative of the vector function \( r(t) \) and evaluate it at that point.
The derivative is given by\(\( r'(t) = \frac{d}{dt} (t \cos(t)i + t \sin(t)j + 2tk) \)\). Evaluating r'(t) at t = 0, we obtain the direction vector of the tangent line. Using the point-direction form of the line equation, we can write the parametric equations for the line tangent to the curve at the given point.
In summary, to show that motion along the curve occurs at an increasing speed as t > 0 increases, we analyze the speed function. To find the parametric equations for the line tangent to the curve at the point\(\((0, \frac{\pi}{2}, \pi)\)\), we differentiate the vector function and evaluate it at that point.
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The elevator at Glen Allen High School has a maximum weight capacity of 2,000 pounds. There are already 3 people on the elevator that have a combined weight of 460 pounds. If 15 people need to get on the elevator, write an inequality expressing the average weight of each additional person.
Answers
Answer:
Therefore, the average weight of each additional person must be less than or equal to 102.67 pounds in order to safely accommodate 15 people on the elevator without exceeding the maximum weight capacity.
Step-by-step explanation:
Let's start by finding the remaining weight capacity of the elevator. Since there are already 3 people on the elevator with a combined weight of 460 pounds, the remaining weight capacity is:
2000 - 460 = 1540 pounds
Now, let x be the average weight of each additional person. To find the maximum number of additional people that can safely get on the elevator, we need to divide the remaining weight capacity by the average weight of each additional person. This gives us the inequality:
x ≤ 1540/15
Simplifying the right-hand side of the inequality, we get:
x ≤ 102.67
Therefore, the average weight of each additional person must be less than or equal to 102.67 pounds in order to safely accommodate 15 people on the elevator without exceeding the maximum weight capacity.
what is the measure of one angle in a regular 24-gon?
Answers
Answer:165degrees
Step-by-step explanation
Use formula N-2 × 180 N is the number of sides
24-2=22
22x180=3960 total
for each angle divide total by 24=165 degrees
in an average quarter hour, the total listening audience in greenville is 35,600. if the radio station clip has an average quarter hour listenership of 8,100, what is its average quarter-hour share?
Answers
Average quarter-hour share is 22.75%
What is meant by Average quarter-hour share?Average quarter-hour share: Average Quarter-Hour Share is the percent of the total listening audience tuned in to each station. This estimate reveals the share of listening each station captures out of the total listening in the survey area.
in an average quarter hour, the total listening audience in green villa is 35,600 and the radio station clip has an average quarter hour listenership of 8,100
we have to find average quarter-hour share
Average quarter hour share = (AQH persons of a station/AQH persons of all station) * 100 = (8100/35600) *100 = 22.75%
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write the ratio in simplest form 1. 30 treadmills to 36 elliptical machines / 2. 180 red marbles to 145 blue marbles / 3. in the word flashlight what is the ratio of vowels to total letters? /
Answers
The ratio in simplest form in the three cases is 5 : 6, 36 : 29 and 1 : 5.
30 treadmills to 36 elliptical machines can be simplified to 5 treadmills to 6 elliptical machines. To simplify, you can divide both numbers by their greatest common factor, which in this case is 6.
30/6 = 5 and 36/6 = 6
So 30 : 36 = 5 : 6
180 red marbles to 145 blue marbles can be simplified to 12 red marbles to 11 blue marbles. To simplify, you can divide both numbers by their greatest common factor, which in this case is 5.
180/5 = 36 and 145/5 = 29
180 : 145 = 36 : 29
In the word "flashlight", the ratio of vowels to total letters is 2:8. The word "flashlight" has 2 vowels (a,i) and 10 total letters. To simplify divide by 2.
2 : 10 = 1 : 5
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1. Jordan is working two summer jobs, making $15
per hour walking dogs and making $25 per hour
clearing tables. In a given week, he can work a
maximum of 32 total hours and must earn at least
$168. If x represents the number of hours walking
dogs and y represents the number of hours clearing
tables, write a system of inequalities that represents
this scenario. The inequalities should have y
isolated.
Answers
The system of inequalities is y ≤ 32 - x and y ≥ (168 - 15x)/25.
Jordan is working for two summer jobs. The amount made per hour by walking dogs is $15. The amount made per hour by clearing tables is $25. Jordan can work for a maximum of 32 hours a week. He must earn at least $168.
The number of hours spent walking dogs is represented by "x". The number of hours spent clearing tables is represented by "y". The inequalities formed are :
x + y ≤ 32
15x + 25y ≥ 168
The graph of the common shaded region is attached below. The solutions lie in the first quadrant, where the region is shaded for both the inequalities.
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List five different vocabulary words that could be used to describe each given expression. A. −+ b. 20−3 c. +2
Answers
Complete Question:
List five different vocabulary words that could be used to describe each given expression.
\((a)\ a - d + c\)
\((b)\ 20 - 3c\)
\((c)\ \frac{b}{d + 2}\)
Answer:
(a) Vocabulary words: sum, difference, total, subtract, add
(b) Vocabulary words: product, difference, subtract, times, less than
(c) Vocabulary words: quotient, sum, add, divide, ratio
Step-by-step explanation:
\((a)\ a - d + c\)
The word problem of this is:
The addition of c to the difference between a and c
Keywords: addition and difference
Vocabulary words: sum, difference, total, subtract, add
\((b)\ 20 - 3c\)
The word problem of this is:
The difference between 20 and the product of 3 and c
Keywords: difference and product
Vocabulary words: product, difference, subtract, times, less than
\((c)\ \frac{b}{d + 2}\)
The word problem of this is:
The quotient of \(b\ and\ the\) sum of d and 2
Keywords: quotient and sum
Vocabulary words: quotient, sum, add, divide, ratio
pls help if ur good at math
Answers
Answer:
1. \(y=-9x+5\)
2. \(y=-3x-7\)
3. \(y=\frac{2}{3} x+2\)
4. \(y=-\frac{1}{4} x-4\)
5. \(y=-\frac{7}{2}x-2\)
6. \(y=-x-9\)
Step-by-step explanation:
What is the probability of NOT drawing a face card from a standard deck of 52 cards.
8 over 13
3 over 13
10 over 13
1 half
Answers
The probability of NOT drawing a face card from a standard deck of 52 cards is 10 over 13.
First determine the total number of face cards and non-face cards in a standard deck of 52 cards. In a standard deck, there are 12 face cards (3 face cards per suit: Jack, Queen, and King, and 4 suits: Hearts, Diamonds, Clubs, and Spades). This means there are 52 - 12 = 40 non-face cards.
Now, we'll calculate the probability of NOT drawing a face card:
Probability = (Number of favorable outcomes) / (Total number of outcomes)
Probability of NOT drawing a face card = (Number of non-face cards) / (Total number of cards)
Probability = 40 / 52
Simplify the fraction by dividing the numerator and denominator by their greatest common divisor (4):
Probability = (40/4) / (52/4)
Probability = 10 / 13
So, the probability of NOT drawing a face card from a standard deck of 52 cards is 10/13. Your answer: 10 over 13.
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plz help meeeee asap
Answers
I hope this helped!!!
Have a great day!
Find the coordinates of the new vertices of the image that has been dilated by a factor of 5 in orderS(-5,2) -->SY(-4,5) -->YR(-1,1) -->RA(-4,-2) -->A
Answers
S( -5, 2) ----------------> S'(-25, 10)
Y(-4, 5) ------------------> Y'(-20, 25)
R(-1, 1) --------------------> R' (-5, 5)
A(-4, -2) -----------------> A' (-20, -10)
Help pls
Tickets at a particular movie theater have different rates for adults and children. On Friday, the theater sold 4 adult tickets and 7 child tickets for $83. The next day, the theater sold 5 adult tickets and 6 children tickets for $90. What is the price for the adult ticket and the price for the child ticket?
Answers
Tickets at a particular movie theater have different rates for an adult ticket is $12 and the price for a child ticket is $5.
what is elimination method?Elimination method is a technique used in solving a system of linear equations. The goal of this method is to eliminate one variable by adding or subtracting two equations in a way that the variable will be eliminated, leaving only one equation with one variable to solve
According to question:Let x be the price for an adult ticket and y be the price for a child ticket. Then we have:
4x + 7y = 83 (equation 1)
5x + 6y = 90 (equation 2)
Either replacement or elimination will work to solve this system.. Here, we'll use elimination:
When we multiply equations 1 and 2 by 5 and by 4, we obtain:
20x + 35y = 415 (equation 1, multiplied by 5)
20x + 24y = 360 (equation 2, multiplied by 4)
Equation 1 is obtained by subtracting equation 2:
11y = 55
Dividing both sides by 11, we get:
y = 5
Substituting y = 5 into equation 1, we get:
4x + 7(5) = 83
Simplifying and solving for x, we get:
4x = 48
x = 12
Therefore, the price for an adult ticket is $12 and the price for a child ticket is $5.
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A rectangular prism has a base that is 5 cm by 7 cm and a height of 12 cm. If all dimensions are doubled, what happens to the volume?
When the dimensions are doubled, the volume is twice the original volume.
When the dimensions are doubled, the volume is three times the original volume.
When the dimensions are doubled, the volume is four times the original volume.
When the dimensions are doubled, the volume is eight times the original volume.
Answers
Answer: When the dimensions are doubled, the volume is eight times the original volume.
Step-by-step explanation:
Volume right now: 420 cm³
If we double all the dimensions, then the new specifications are 10, 14, and 24 inches respectively. This means that the new volume will be 3360cm³.
The relationship between the new volume and the old volume are: 3360/420 = 8
So the new volume is 8 times the original volume. Your answer is the last option
Answer:
The answer is D
Step-by-step explanation:
Present and future value tables of $1 at 9% are presented below. Esquire Company will need to update some of its manufacturing equipment in the future. In order to accumulate the necessary funds, Esquire will deposit \$5,800into a money market fund at the end of each year for the next six years. How much will accumulate by the end of the sixth and final payment if the fund earns 9% interest compounded annully? Multiple Choice $37,410 $43,635 $37,410 $43,635 $37,932
Answers
The amount that will accumulate by the end of the sixth and final payment is approximately $41,666.60.
To calculate the accumulated amount by the end of the sixth and final payment, we can use the future value of an ordinary annuity formula:
Future Value = Payment × Future Value of an Ordinary Annuity Factor
The payment is $5,800, and the interest rate is 9%. Since the payments are made at the end of each year, we can use the future value table for an ordinary annuity at 9%.
Looking up the factor for 6 years at 9% in the future value table, we find it to be 7.169858.
Now we can calculate the accumulated amount:
Future Value = $5,800 × 7.169858 = $41,666.60
Therefore, the amount that will accumulate by the end of the sixth and final payment is approximately $41,666.60. The correct answer is not among the options provided.
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Write the equation of the line that passes through the points (1, -1) and (6, -1). Put your answer in fully simplified point-slope form, unless it is a vertical or horizontal line.
Answers
The equation of the line that passes through the points (1, -1) and (6, -1) y = - 1 which is a horizontal line.
Given that:
Points, (1, -1) and (6, -1)
Let the equation of the line pass through (x₁, y₁) and (x₂, y₂). Then the equation of the line is given as,
\(\rm (y - y_2) = \left (\dfrac{y_2 - y_1}{x_2 - x_1} \right ) (x - x_2)\)
The equation of the line that passes through the points (1, -1) and (6, -1) is calculated as,
(y + 1) = [(- 1 + 1) / (6 - 1)] (x + 1)
y + 1 = 0
y = - 1
The equation of the line that passes through the points (1, -1) and (6, -1) y = - 1 which is a horizontal line.
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Subtract -5x ^ 2 - 9x + 8 from - 6x ^ 2 + 5
Answers
Answer:
x² - 9 x +3
Step-by-step explanation:
(-5x²)-(-6x²) = 1x²
8 - 5 = 3
The probability of getting a sum of 7 or 11 by rolling two dice is 2/9 True or false please answer my question
Answers
Answer:
IT IS FALSE
Step-by-step explanation:
How do I convert 6-2y=-x to y=Mx+b?
Answers
6-2y= -x
subtract 6 on both sides
Now it should look like -2y= -x - 6
Now divide everything by -2
Now it should look like y= -x -6
Hope this helps sorry if it’s confusing
event a happens 50% of the time, event B happens 50% of the time, what is the probability they overalp
Answers
The likelihood of events A and B overlapping is 25% (or 0.25), assuming both are unrelated. To determine the precise likelihood, extra information is required if they are dependent.
Whether occurrences A and B are independent or dependent determines the likelihood that they will overlap. Let's think about both possibilities:
Events A and B must be independent of one another for their probabilities to be unaffected by one another.
In this scenario, there is a 50% (0.5) chance of event A happening and a 50% (0.5) chance of event B happening.
We add their probabilities together to get the likelihood that the two occurrences will occur simultaneously:
Probability (A and B) = Probability (A) * Probability (B) = 0.25 or 0.5%.Events A and B that are dependent on one another may have different effects on their probabilities.
It is difficult to calculate the exact probability of overlap without additional knowledge about the dependency's nature.
the likelihood of events A and B overlapping is 25% (or 0.25), assuming both are unrelated.
To determine the precise likelihood, extra information is required if they are dependent.
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Find Zα/2 for α = 0.19. Click the icon to view the table of normal curve areas. Zα/2 = ____ (Round to two decimal places as needed.)
Answers
Zα/2 for α = 0.19, we need to refer to the standard normal distribution table, so using this, Zα/2 = 1.65.
To locate Zα/2 for α = 0.19, we need to refer to the usual normal distribution table.
Since the price of α/2 is 0.19/2 = 0.0.5, we want to find the corresponding area under the everyday curve this is closest to 0.Half.
Using the everyday distribution table, we are able to locate the closest cost to zero.0.5, which is 0.0968. The corresponding Z-score for this area is about 1.65.
Therefore, Zα/2 = 1.65 (rounded to 2 decimal places).
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Which function shown below has the greater rate of change? Explain. (Be sure to include the
rate of change for each in your explanation).
Answers
Answer:
Function 1 has a greater rate of change.
Step-by-step explanation:
Function 1 has a rate of change of 3 because 3x is the slope of the line.
Function 2 has a rate of change of 1.5, because when you use rise/run, you get a slope of 3/2.
Function 3 has a rate of change of 2. You use the formula (y2 - y1) / (x2 - x1) to find the slope. I used (0,0) and (1,2). When you plug in 2 for y2, 1 for x2, and 0 for y1 and 0 for x1, you get (2-0) / (1-0) which is 2/1. That equals a slope of 2.