The series ∑ₙ=₁ to ∞ (n/n² + 6) is divergent.
To determine whether the series ∑ₙ=₁ to ∞ (n/n² + 6) is convergent or divergent, we can use the Integral Test.
The Integral Test states that if f(x) is a continuous, positive, and decreasing function on the interval [1, ∞) and f(n) = aₙ for all positive integers n, then the series ∑ₙ=₁ to ∞ aₙ and the integral ∫₁ to ∞ f(x) dx either both converge or both diverge.
In this case, let's consider the function f(x) = x/(x² + 6). We can check if it meets the conditions of the Integral Test.
Positivity: The function f(x) = x/(x² + 6) is positive for all x ≥ 1.
Continuity: The function f(x) = x/(x² + 6) is a rational function and is continuous for all x ≥ 1.
Decreasing: To check if the function is decreasing, we can take the derivative and analyze its sign:
f'(x) = (x² + 6 - x(2x))/(x² + 6)² = (6 - x²)/(x² + 6)²
The derivative is negative for all x ≥ 1, which means that f(x) is a decreasing function on the interval [1, ∞).
Since the function f(x) = x/(x² + 6) satisfies the conditions of the Integral Test, we can evaluate the integral to determine if it converges or diverges:
∫₁ to ∞ x/(x² + 6) dx
To evaluate this integral, we can perform a substitution:
Let u = x² + 6, then du = 2x dx
Substituting these values, we have:
(1/2) ∫₁ to ∞ du/u
Taking the integral:
(1/2) ln|u| evaluated from 1 to ∞
= (1/2) ln|∞| - (1/2) ln|1|
= (1/2) (∞) - (1/2) (0)
= ∞
The integral ∫₁ to ∞ x/(x² + 6) dx diverges since it evaluates to ∞.
According to the Integral Test, since the integral diverges, the series ∑ₙ=₁ to ∞ (n/n² + 6) also diverges.
Therefore, the series ∑ₙ=₁ to ∞ (n/n² + 6) is divergent.
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Incomplete question:
Use the Integral Test to determine whether the series is convergent or divergent.
∑ₙ=₁ to ∞ = n/n² + 6
Evaluate the following integral ∫₁ to ∞ x/x²+6 . dx
Write an equation for the following scenario.
A 4000 fish population of tuna has been growing continuously at a rate of 0.3%.
[tex]4000(1.003)^{(x-1)}[/tex]
Step-by-step explanation:
Michelle wants to buy a new phone. There are several stores that sell the phone she wants, and each store offers
a different discount. The information on each store's discount offer for the phone Michelle wants to buy is
summarized in the table.
Price
Discount
Store
Big Town Electronics
$120
10% off
Central Phones
$120
$10 off
Main Street Mobile
$160
30% off
Phone Forum
$125
$16 off
Wireless Inc.
$110
No discount
At which store would Michelle's total cost be the least after the discount?
А
Big Town Electronics
B
Wireless Inc.
с
Main Street Mobile
D
Phone Forum
Step-by-step explanation:
hey, hope it helps.. :))
What is the equation of the axis of symmetry?
help PLz will mark brain list
Answer:
10.10 15.15
Step-by-step explanation:
25.30 15.12 25.13
Find the volume of the square pyramid. The work has been started for you.
V = 1
3
Bh
B = 6.5(6.5) = 42.25 m2
Complete question
A square pyramid. The square base has side lengths of 6.5 meters and the height of 6 meters.
What is the volume of the pyramid?
Answer:
84.5m³
Step-by-step explanation:
The volume of a square. Pyramid is given as :
V = 1/3 * base * height
Base = l² ; l = side length
Base = 6.5² = 42.25 m²
V = 1/3 * 42.25 * 6
V = 42.25 * 2
V = 84.5
The volume of square pyramid = 84.5m³
Write and solve an equation to find the missing dimension of the figure.
(WILL GIVE BRAINLYSSS)
Answer:
2(11*9) + 2(11*h) + 2(9*h) = 558
h = 9 in.
Step-by-step explanation:
2(11*9) + 2(11*h) + 2(9*h) = 558
198 + 22h + 18h = 558
40h = 558-198
40h = 360
h = 9 in.
In a survey, music club members select their preference between Song A or Song B. Song A is selected by 30 teens and 10 adults. Of 20 members who select Song B, five are teens.
Make a two-way frequency table to organize the data.
Is it reasonable to say that more people surveyed prefer Song A? Explain.
Is it reasonable to say that more adults than teens participated in the survey? Explain.
Answer: 40 people like song A whereas 20 people like song B. Thus, it is reasonable to say that more people liked Song A .
Step-by-step explanation:
Will help you with something if u help me plsss
Steph Curry went crazy buying houses and carsafter the Warriors won the first round of the playoffs. Now he doesn’t have enough money to make his mortgage payment. Alyssa and Step are good friends, so Alyssa decided to lend Steph $7,500. Alyssa is going to charge Step 8% interest, which will compound monthly. How much will Steph owe Alyssa if he pays her back in 2 years?
Answer:
21900
Step-by-step explanation:
Find 8% of 7500 (600) and multiply it by 24 (amount of months) then add that to 7500
How many cups are in 20 gallons? AND how do I show my work for that?
correct=brainliest
We are making two fruit drinks, Red berry (R) and Green Mush (GM). The drinks contain a combination of cherry juice (C), cranberry juice (CB) and avocado (A). Red Berry sells for $9 a gallon and Green Mush sells for $11 a gallon. We need at least 100 gallons of red berry and 50 gallons of green mush. Cherry juice contains 400 units vitamin C per gallon, cranberry juice contains 350 units of vitamin C per gallon and avocado contains 200 units of vitamin C. Cherry juice costs $2 per gallon, cranberry juice $1.50, and avocado costs $5. Red Berry must contain at least 325 units of vitamin C per gallon. Green Mush must contain a minimum of 150 units of vitamin C. We have 50 gallons of cherry juice, 70 gallons of cranberry juice and unlimited supply of avocado juice.
The objective function is
One decimal place examples 4.0 or 4.1
Z =
______________ XC,RB+
_______________XCB,RB+
________________XA,RB+
________________XC,GM+
_________________XCB,GM+
___________________XA,GM
The constrint for minimum vitamin C for Red Berry is
No decimal places example 4 negatives as -4 not parenthesis
______________ XC,RB+
_______________XCB,RB+
________________XA,RB+0XC,GM+0XCB,GM+0XA,GM <=
____________________
Objective function is Z = 9 XC,RB + 11 XCB,GM, and the constraint for minimum vitamin C for Red Berry is75XC,RB + 350XCB,RB + 200XA,RB >= 0.
Objective function for the given statement is Z = 9 XC,RB + 11 XCB,GM,
where, XC, RB is the number of gallons of Cherry juice used in Red Berry, XCB, GM is the number of gallons of Cranberry juice used in Green Mush and also, XA, RB is the number of gallons of Avocado juice used in Red Berry, XC, GM is the number of gallons of Cherry juice used in Green Mush, XCB, GM is the number of gallons of Cranberry juice used in Green Mush, XA, GM is the number of gallons of Avocado juice used in Green Mush.
Hence, the objective function is Z = 9 XC,RB + 11 XCB,GM.
Minimum vitamin C for Red Berry will be given by the equation,
350XCB,RB + 400XC,RB + 200XA,RB >= 325XC,RB
=> 75XC,RB + 350XCB,RB + 200XA,RB >= 0
So, the constraint for minimum vitamin C for Red Berry is75XC,RB + 350XCB,RB + 200XA,RB >= 0.
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I can't figure it out
Answer:
V = 17,371.8
Step-by-step explanation:
V = π r^2 h
V = π (12)^2 (38.4)
V = 17,371.75073729
V = 17,371.8
You have to calculate the area of the base of the cylinder (the circle) by using the formula π r^2, r = the radius of the circle. You are squaring the r because r^2 = diameter. The formula to get the area of a circle is π r ^2. To get the volume you need to multiply the area of the circle by the height of the cylinder. All together the total formula to get the volume of a cylinder is π r^2 h
If x=(y+2)^2 and y= -7 then what is x
Answer:
X=25
Step-by-step explanation:
y+2=-7+2=5
[tex]5^{2} =25[/tex]
What is equity? What does it mean to have positive equity? What does it mean to have negative equity?
Last week's and this week's low temperatures are shown in the table below.
Low Temperatures for 5 Days This Week and Last Week
Low Temperatures.
This Week (°F)
Low Temperatures
Last Week (°F)
4
10
13 9
6
5
9
8
6
LO
5
Which measures of center or variability are greater than 5 degrees? Select three choices.
the mean of this week's temperatures
O the mean of last week's temperatures
the range of this week's temperatures
the mean absolute deviation of this week's temperatures
the mean absolute deviation of last week's temperatures
The mean absolute deviation of last week's temperatures is 7.2°F.
Given below is the table of last week's and this week's low temperatures:Last week's temperatures: 29°F, 35°F, 42°F, 46°F, 52°FThis week's temperatures: 27°F, 31°F, 35°F, 38°F, 42°F, 46°F, 50°F
The range of this week's temperatures is found by subtracting the smallest value from the largest value. Therefore, the range of this week's temperatures is 50°F - 27°F = 23°F.
Mean absolute deviation is a measure of how much the data deviates from the mean of the data. To find the mean absolute deviation of last week's temperatures, we first need to find the mean of the data set.Using the formula for mean, we have:
Mean = (29 + 35 + 42 + 46 + 52)/5 = 40.8°F
To find the absolute deviations of each temperature from the mean, we need to subtract each temperature from the mean and take the absolute value.Absolute deviations:
|29 - 40.8| = 11.8|35 - 40.8|
= 5.8|42 - 40.8|
= 1.2|46 - 40.8|
= 5.2|52 - 40.8|
= 11.2
Next, we need to find the mean of these absolute deviations. Using the formula for mean again, we have:Mean absolute deviation = (11.8 + 5.8 + 1.2 + 5.2 + 11.2)/5 = 7.2°F
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We've established that heights of 10-year-old boys vary
according to a Normal distribution with mu = 138cm and sigma = 7cm
What proportion is between 152 and 124 cm ?
The proportion of values between 152 and 124 cm is 0.8996 or 89.96%.
Proportion between 152 and 124 cm according to a normal distribution with mu = 138cm and sigma = 7cm is 0.8996.
According to the given question, we know that the mean is μ = 138 cm, standard deviation is σ = 7 cm.
We have to find the proportion that is between 152 and 124 cm.
To solve the problem, first we have to find the z-scores for 152 and 124 cm.
We can calculate the z-scores as follows:Z-score for 152 cm is given by:z152=(152−138)7=2z_{152}=\frac{(152-138)}{7}=2z152=(152−138)7=2Z-score for 124 cm is given by:z124=(124−138)7=−2z_{124}=\frac{(124-138)}{7}=-2z124=(124−138)7=−2
We can use a z-table or calculator to find the proportion of values between these two z-scores.
The area under the standard normal curve between z = -2 and z = 2 is approximately 0.8996.
Therefore, the proportion of values between 152 and 124 cm is 0.8996 or 89.96%.
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Which of the following describes the root of the following function? f[x] = -x2 3x + 1 Select one a. Exactly 1 rational root. b. 2 distinct rational roots. c. 2 distinct irrational roots. d. 2 distinct imaginary roots.
Answer:
please stop being lazy by using OCR! next time type the question!
there's nothing such as = -x2 3x + 1
it's probablhy f(x) = -x² + 3x + 1
Answer:
2 distinct irrational roots
Step-by-step explanation:
The solution set of IR - 31 = 10 is?
{7, 13)
(-7, 13)
(-13, 13)
Answer:
{7, 13)
Step-by-step explanation:
IR - 31 = 10
IR = 41
7*13 = 41
(-7)*13 = -41
(-13)*13 = -169
(1 point) suppose that the trace of a 2×2 matrix a is tr(a)=12 and the determinant is det(a)=32. find the eigenvalues of a. the eigenvalues of a are . (enter your answers as a comma separated list.)
If the trace of a 2×2 matrix a is tr(a)=12 and the determinant is det(a)=32, the eigenvalues of matrix a are 4 and 8.
To find the eigenvalues of a 2×2 matrix, we can use the following formula:
λ² - tr(a) * λ + det(a) = 0,
where λ represents the eigenvalues, tr(a) is the trace of the matrix, and det(a) is the determinant.
In this case, we are given that tr(a) = 12 and det(a) = 32. Substituting these values into the formula, we have:
λ^2 - 12λ + 32 = 0.
To solve this quadratic equation, we can factorize it or use the quadratic formula. Factoring the equation, we get:
(λ - 4)(λ - 8) = 0.
Setting each factor equal to zero, we find two solutions:
λ - 4 = 0 => λ = 4,
λ - 8 = 0 => λ = 8.
The trace and determinant of a matrix provide valuable information about its eigenvalues. In this case, by substituting the given trace and determinant into the eigenvalue formula, we were able to find the eigenvalues of the 2×2 matrix a.
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Steam enters an inclined pipe with an elevation change of +71 m operating at steady state with a specific enthalpy of 2744 kJ/kg and a mass flow rate of 3 kg/s. Assuming there is no significant change in kinetic energy from inlet to exit, and the rate of heat transfer from the surrounding to steam is 696 kW, what is the specific enthalpy at the exit of the pipe?
The specific enthalpy at the exit of the pipe is 2804 kJ/kg.
The expression of specific enthalpy at the exit of the pipe can be found by the following method:
Here, m = mass flow rate of the steam = 3 kg/s
q1 = specific enthalpy at inlet of the pipe = 2744 kJ/kg
W = work done on the system = 0 (steady-state)
Q = rate of heat transfer = 696 kW
z1 = elevation of the inlet = 0 m
z2 = elevation of the outlet = +71 m
Since the rate of heat transfer is given as
Q = -m(h2 - h1) + W + Q_hot
So, h2 = [Q - Q_hot + m(h1) - W]/m
Where W = 0 and Q_hot = 0
h2 = (696 - 0 + 3(2744))/3
h2 = 2804 kJ/kg
Therefore, the specific enthalpy at the exit of the pipe is 2804 kJ/kg.
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Find the marginal distribution of military status in percentages. Round to the nearest whole percent.
The question is incomplete. Here is the complete question.
Mavis is interested whether military status is related to gender in American Samoa. The data she collected about the population of American Samoa in 2010 is below.
Find the marginal distribution of military status in percentages. Round to the nearest whole percentages.
In Armed Forces:
Military-dependent:
Civilian:
Answer: In Armed Forces: 0%
Military-dependent: 3%
Civilian: 97%
Step-by-step explanation: Marginal Distribution is defined as the probability distribution of variables contained in a subset.
For the American Samoa's military status, marginal distribution is
1) In Armed Forces:
[tex]p_{1}=\frac{87}{55,519}*100[/tex]
[tex]p_{1}=0.15[/tex]
2) Military-dependent:
[tex]p_{2}=\frac{1709}{55,519}*100[/tex]
[tex]p_{2}=3.08[/tex]
3) Civilian:
[tex]p_{3}=\frac{53,723}{55,519}*100[/tex]
[tex]p_{3}=96.77[/tex]
Now round each percentage to the nearest whole percentage:
Armed forces: 0%
Military: 3%
Civilian: 97%
The marginal distributions of military status are, in nearest whole percentage:
Armed forces: 0%
Military: 3%
Civilian: 97%
The area of a circular rose garden is 88
square meters.
What is the radius of the garden?
Answer:
c
Step-by-step explanation:
PLEASE ANSWER MY RECENT! ITS FOR 47 POINTS AND SUPER EASY! I JUST NEED ADVICE AND NO ONE HAS ANSWERD YET!
Answer:
Hi
Step-by-step explanation:
Problem #1: Ms. Barrett is doing an art project with her class. She has 4 sheets of tissue paper. If she gives each student a third of a sheet, how many students will get tissue paper?
Answer:
12 students.
Step-by-step explanation:
If you had 4 sheets, you would want to start by dividing them all in 3's, since that's easiest. (open the file i just for a better example of how the sheet should look).
Now, let's assume these sheets are regular office sized paper.
Start by handing 1 third out to A student.
Now, since there are 4 sheets, divided in 3 sections, we multiply to see the result of how many sections there are.
4 x 3 = 12.
Therefore, 12 students will receive a third of the tissue paper.
Which of these statements is true? Check all that apply.
CDs have a higher mean than digital.
The range of digital is 800.
The median of CDs is 400,
Both have the same interquartile range.
O Both have the same median,
Digital's mean is around 467.
Answer:
abd
Step-by-step explanation:
im smart
Answer:
The answer is A B D F
Step-by-step explanation:
I took the test sorry I don't have a better explanation.
How do you find the value of x while working with adjacent and vertical angles? The angle is adjacent btw if that helps... I am in 7th grade and need help with this topic
Answer:
The angles are adjacent because they have a common side and vertex. Basically, they are both right next to each other.
x = 28
Step-by-step explanation:
When (x + 6)° and 56° are added together, they need to equal 90° since they are both parts of a right angle.
(x + 6) + 56 = 90
x + 62 = 90
x = 28
Ken jumps 1 1/5 metres steve jumps 1.5 metres steve jumps further than ken how much further does steve jump than Ken?
Answer:0.3 m
Step-by-step explanation:
Given
Ken can Jump [tex]1\ \frac{1}{5}\ m[/tex]
Steve can Jump [tex]1.5\ m[/tex]
Converting mixed fraction into a fraction
[tex]\Rightarrow 1\ \dfrac{1}{5}=\dfrac{1\times 5+1}{5}\\\\\Rightarrow \dfrac{6}{5}\ m=1.2\ m[/tex]
The difference between their Jumps is
[tex]\Rightarrow 1.5-1.2=0.3\ m[/tex]
Steve Jumps 0.3 m more than ken
What is the arc length the car traveled to the nearest hundredth?
a. 7.91
b. 8.32
c. 10.99
d. 11.89
To find the arc length, we can use the formula:
[tex]\[ L = \frac{\theta}{360} \times 2\pi r \]\\Given:\( r = \frac{d}{2} = \frac{30}{2} = 15 \) ft\( \theta = 42 \) degrees\( \pi = 3.14 \)\\Substituting the given values into the formula:\[ L = \frac{42}{360} \times 2 \times 3.14 \times 15 \]\[ L = \frac{42}{360} \times 94.2 \]\[ L = \frac{3956.4}{360} \]\[ L \approx 10.99 \] ftTherefore, the arc length traveled by the car, to the nearest hundredth, is 10.99 feet. The correct option is c.[/tex]
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What does it mean if two equations have no solution ?
Answer:
this means there is no answer to the problem
Step-by-step explanation:
How many different 3-digit odd number can be formed with the digit 0,1,2 up to 9 if the repitition of digits is not allowed
Answer:
The answer is "320".
Step-by-step explanation:
They have to be 1, 3, 5, 7, and 9 as your final digit. Your middle digit may be anything other than the first digit, and only the last digit or 0, is anything.
Now let us take a look at strange digits ending in 1.
This can start from any digit other than 0, so there are 8 options during the first digit.
Its middle digit could only be 1 digit and 1 digit. Even so, it can also be 0, since it has 8 options.
So, [tex]8\times 8 = 64[/tex] strange number ending in 1.
For other strange amounts, the very same logic is working.
And thus there are 64 possible odd numbers for just about any number ending in any of the five odd digits: 1, 3, 5, 7, or 9.
The answer is [tex]5 \times 64 = 320[/tex].
7
If n=100 and p (p-hat) = 0.72, construct a 95% confidence interval. Give your answers to three decimals.
To find the 95% confidence interval when n=100 and p=0.72, we can use the following formula:
$$\left(\hat{p}-z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, \hat{p}+z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\right)$$
Where, $\hat{p}$ is the point estimate of the population proportion, $n$ is the sample size, $z_{\alpha/2}$ is the critical value of the standard normal distribution at a significance level of $\alpha$, which can be obtained from a table. For a 95% confidence level, $\alpha$ is equal to 0.05/2 = 0.025 on each tail.
The corresponding z-value is 1.96 (approximately).Hence, plugging in the values, we get
$$\begin{aligned}\left(0.72-1.96 \sqrt{\frac{0.72(0.28)}{100}}, 0.72+1.96 \sqrt{\frac{0.72(0.28)}{100}}\right) \\ \left(0.631, 0.809\right)\end{aligned}$$
Therefore, the 95% confidence interval is (0.631, 0.809) rounded to three decimal places.
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