can someone help me with this question? Thank you!

Best guess for the function is

\(\displaystyle f(x) = \sum_{n=1}^\infty \frac{x^n}{n^2}\)

By the ratio test, the series converges for

\(\displaystyle \lim_{n\to\infty} \left|\frac{x^{n+1}}{(n+1)^2} \cdot \frac{n^2}{x^n}\right| = |x| \lim_{n\to\infty} \frac{n^2}{(n+1)^2} = |x| < 1\)

When \(x=1\), \(f(x)\) is a convergent \(p\)-series.

When \(x=-1\), \(f(x)\) is a convergent alternating series.

So, the interval of convergence for \(f(x)\) is the closed interval \(\boxed{-1 \le x \le 1}\).

The derivative of \(f\) is the series

\(\displaystyle f'(x) = \sum_{n=1}^\infty \frac{nx^{n-1}}{n^2} = \frac1x \sum_{n=1}^\infty \frac{x^n}n\)

which also converges for \(|x|<1\) by the ratio test:

\(\displaystyle \lim_{n\to\infty} \left|\frac{x^{n+1}}{n+1} \cdot \frac n{x^n}\right| = |x| \lim_{n\to\infty} \frac{n}{n+1} = |x| < 1\)

When \(x=1\), \(f'(x)\) becomes the divergent harmonic series.

When \(x=-1\), \(f'(x)\) is a convergent alternating series.

The interval of convergence for \(f'(x)\) is then the closed-open interval \(\boxed{-1 \le x < 1}\).

Differentiating \(f\) once more gives the series

\(\displaystyle f''(x) = \sum_{n=1}^\infty \frac{n(n-1)x^{n-2}}{n^2} = \frac1{x^2} \sum_{n=1}^\infty \frac{(n-1)x^n}{n} = \frac1{x^2} \left(\sum_{n=1}^\infty x^n - \sum_{n=1}^\infty \frac{x^n}n\right)\)

The first series is geometric and converges for \(|x|<1\), endpoints not included.

The second series is \(f'(x)\), which we know converges for \(-1\le x<1\).

Putting these intervals together, we see that \(f''(x)\) converges only on the open interval \(\boxed{-1 < x < 1}\).

Answer:

slope intercept form is mx+b=y where the slope is 2 because 6/3=2 so therefore we need to find b which you can subsitute the values to get 2(2)+b=(0) so b=-4 therefore the slope intercept is 2x-4=y

to check my answer we used (2,0) so we use (5,6) to get 2*5-4=6 which is a true statement

give me brainliest :)

Step-by-step explanation:

If attendance has climbed by 6% annually since 2015, we can estimate that 34,269 people will attend the festival in 2021.

We can use the formula for exponential growth:

\(y = a * (1 + r)^x\)

Where

Plugging in the values, we get:

\(y = 25,000 * (1 + 0.06)^x\)

Simplifying, we get:

\(y = 25,000 * 1.06^x\)

As 2021 is six years from 2015, we need to know the attendance at x = 6 in order to estimate how many people will attend the event in 2021. When we enter x = 6, we obtain:

\(y = 25,000 * 1.06^6\)

\(y =34,268.77\)

Therefore, If attendance has climbed by 6% annually since 2015, we can estimate that 34,269 people will attend the festival in 2021.

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Answer:

$4.50

Step-by-step explanation:

You would divide 6 by 4 to get 1.5 being $1.50 and then muiltpy it by 3 to get 4.50.

Answer:

180 - 119.4 = 60.6

So the answer is 60.6

Answer:

60.6°

Step-by-step explanation:

Supplementary angles equal 180 degrees. So we can show this with the following equation:

119.4 + x = 180

where x is the measure of the other supplementary angle.

If we subtract 119.4 from both sides, we get the the value of x, which is 60.6.

Answer:

2,075 = 5 × 5 × 83

√2,075 = 5√83 = about 45.55

Answer: Area = 126 square inches

Explanation:

The formula for calculating the area of a triangle is expressed as

Area = 1/2 x base x height

From the information in the diagram,

base = 18

height = 14

Thus,

Area = 1/2 x 18 x 14

Area = 1/2 x 252 = 252/2

Area = 126 square inches

Answer:

2y -2 = 3x +9

or

3x -2y = -11

Answer:

3x - 2y = - 11

Step-by-step explanation:

Ax + By = C

A, B and C are integers

~~~~~~~~~~~~~~~

y - 1 = 1.5 ( x + 3 )

2y - 2 = 3 ( x + 3 )

3x - 2y = - 11

yes, because the midpoints (a+c) = b

Given that the line's equation is y=x.

B and C are the two points on this line.

( a+ c, b) is the midpoint of B and C.

We can see from this that using the midpoint formula

a + c = \(\frac{x1+x2}{2}\)

b=\(\frac{y1+y2}{2}\)

Because both points are on y=x, we get

x1=y1, x2=y2.

As a result, x1 = a+c and 2y1 = 2b or y1 =b.

Because x1=y1, we have a+c =b.

In other words, the coordinates of the midpoint satisfy the equation y=x.

Thus, the answer is

because (a+c) = b

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Answer:

hope this correct -1/4...............

Step-by-step explanation:

Circumference =

\(\pi r{}^{2} \)

So, we can take the value of as 22/7

\(\pi\)

"r" stands for radius. Radius is half the diameter.

So Radius =120/2=60

Now,

22/7*60*60=11314. 28cm2

PLEASE APPRECIATE THE EFFORT AND MARK AS BRAINILIEST.

Mike's bonus for the year is between $15,000 and $19,500.

The inequality that best represents Mike's bonus, B, for the year is:

$15,000 \(\leq B \leq\) 19,500$

to see why, we are able to use the given data that Mike's bonus is calculated as 3 percent of his corporation's overall profits.

If we let P be the organization's general income, then Mike's bonus B can be expressed as:

$B = 0.03P$

We recognise that the organization's total profits are between $500,000 and $650,000, so we will write:

$500,000 \(\leq P \leq\) 650,000$

Substituting this inequality into the equation for Mike's bonus, we get:

$15,000 \(\leq B \leq\) 19,500$

Therefore, Mike's bonus for the year is between $15,000 and $19,500.

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Answer:

60

Step-by-step explanation:

the bird is flying 40 meter above sea level and the fish is 20 meters below the sea level

I don't know exactly what you're asking but 80% of 100 is 80

Answer:

0.83

Step-by-step explanation:

It’s right use the formula they give you

The probability of a student taking the bus given that the student is sophomore is 0.83

Given that:

P(Walk to school | Sophomore) = 3 /(2 + 25 + 3) = 0.1

To cal culate the probability of a student taking the bus given that the student is sophomore, we make use of:

P(Take the bus | Sophomore ) = 25/(2 + 25 + 3)

Rewrite properly as:

\(Pr= \frac{25}{2 + 25 + 3}\)

So, we have:

\(Pr= \frac{25}{30}\)

Divide 25 by 30

\(Pr= 0.83\)

Hence, the probability of a student taking the bus given that the student is sophomore is 0.83

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Answer:

176 children and 105 adults.

Explanation:

Let's call x the number of children and y the number of adults.

If 281 people entered the park, we can write the following equation

x + y = 281

If they collected 684 dollars, we can write the following equation

1.5x + 4y = 684

because it cost $1.5 for children and $4 for adults.

Now, we have the following system of equations

x + y = 281

1.5x + 4y = 684

First, we need to solve the first equation for y, so

x + y = 281

x + y - x = 281 - x

y = 281 - x

Then, replace this expression on the second equation

1.5x + 4y = 684

1.5x + 4(281 - x) = 684

1.5x + 4(281) - 4(x) = 684

1.5x + 1124 - 4x = 684

-2.5x + 1124 = 684

Finally, we can solve the equation for x

-2.5x + 1124 - 1124 = 684 - 1124

-2.5x = -440

-2.5x/(-2.5) = -440/(-2.5)

x = 176

So, the value of y is equal to

y = 281 - x

y = 281 - 176

y = 105

Therefore, they admitted 176 children and 105 adults.

The correct answer is 113. To forecast the sales of the ice rink on day 4 using the trend method, we need to determine the trend equation based on the given data points.

The trend equation represents the overall pattern or trend in the sales data and allows us to make predictions for future values.

First, we need to calculate the average increase in sales per day. The average increase is obtained by dividing the total increase in sales over the three days (102 - 80 = 22) by the number of days (3 - 1 = 2). Therefore, the average increase in sales per day is 22 / 2 = 11.

Next, we can use the average increase to forecast the sales for day 4. Starting from the last known sales value (102), we add the average increase to project the sales for the next day. Thus, the forecasted sales for day 4 would be 102 + 11 = 113.

Therefore, the correct answer is 113.

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Answer:

ok

Step-by-step explanation:

ok

Therefore, we have shown that 4n is a solution to the recurrence relation an = 3an-1 + 4an-2.

To show that 4n is a solution to the recurrence relation an = 3an-1 + 4an-2, we need to substitute 4n into the recurrence relation and verify that it satisfies the equation for all values of n.

Let's substitute 4n into the recurrence relation:

a_n = 3a_n-1 + 4a_n-2

Replacing a_n with 4n:

4n = 3(4n-1) + 4(4n-2)

Simplifying the right side of the equation:

4n = 3(4n-1) + 4(4n-2)

4n = 3 * 4n-1 + 16n-2

4n = 12n-3 + 16n-2

4n = 12n-3 + 16/n^2

Now, let's simplify the equation further:

4n = 12n-3 + 16/n^2

To manipulate this equation, we need to rewrite the right side so that it has a common denominator:

4n = (12n^3 + 16) / n^2

To make the denominator of the right side equal to n^2, we can multiply the numerator by n^2:

4n = (12n^3 + 16) / n^2

4n = (12n^3n^2 + 16) / n^2

4n = (12n^5 + 16) / n^2

Simplifying further:

4n = 12n^5 / n^2 + 16 / n^2

4n = 12n^3 + 16 / n^2

We can observe that the equation holds true for all values of n, as both sides are equal.

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Answer:

EZ 12678906543234567

Step-by-step explanation:

TOLD YOU :D

BRAINLY PLZ

8X4 =32

8+4 IS 12

SO I DONT KNOW BUT YA

Answer:

(D)Perpendicular lines intersect to form right angles and congruent segments do not have equal length: false

Explanation:

Therefore, the compound statement P^~S will be:

P^~S: Perpendicular lines intersect to form right angles and congruent segments do not have equal length.

Now, the negation of S is False, therefore: P^~S is False.

Answer:

25 Ounces =

1.5625 Pints

or

25 Ounces =

25 Pints

16

Step-by-step explanation:

Answer:

Step-by-step explanation:

1 pint is bigger because 16 fluid ounces would equal to 1 pint so 1 pint is greater

Answer:

v=20π ft/s

Step-by-step explanation:

Given:

Distance from the security car to the movie theater, D=25 ft

Distance of the reflection from the car, d=30 ft

Speed of rotation of the strobe lights, 2 rev/s

To find the speed at which the reflection of the security strobe lights is moving along the wall of the movie theater, we need to calculate the linear velocity of the reflection when it is 30 ft from the car.

We can start by finding the angular velocity in radians per second. Since the strobe lights rotate at 2 revolutions per second, we can convert this to radians per second.

ω=2πf

=> ω=2π(2)

=> ω=4π rad/s

The distance between the security car and the reflection on the wall of the theater is...

r=30-25= 5 ft

The speed of reflection is given as (this is the linear velocity)...

v=ωr

Plug our know values into the equation.

v=ωr

=> v=(4π)(5)

∴ v=20π ft/s

Thus, the problem is solved.

The speed of the reflection of the security strobe lights along the wall of the movie theater is 2π ft/s.

To solve this problem, we can use the concept of related rates. Let's consider the following variables:

x: Distance between the security car and the movie theater wall

y: Distance between the reflection of the security strobe lights and the security car

θ: Angle between the line connecting the security car and the movie theater wall and the line connecting the security car and the reflection of the strobe lights

We are given:

x = 25 ft (constant)

y = 30 ft (changing)

θ = 2 revolutions per second (constant)

We need to find the speed at which the reflection of the security strobe lights is moving along the wall (dy/dt) when the reflection is 30 ft from the car.

Since we have a right triangle formed by the security car, the movie theater wall, and the reflection of the strobe lights, we can use the Pythagorean theorem:

x^2 + y^2 = z^2

Differentiating both sides of the equation with respect to time (t), we get:

2x(dx/dt) + 2y(dy/dt) = 2z(dz/dt)

Since x is constant, dx/dt = 0. Also, dz/dt is the rate at which the angle θ is changing, which is given as 2 revolutions per second.

Plugging in the known values, we have:

2(25)(0) + 2(30)(dy/dt) = 2(30)(2π)

Simplifying the equation, we find:

60(dy/dt) = 120π

Dividing both sides by 60, we get:

dy/dt = 2π ft/s

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A point estimate is the value of a statistic that estimates the value of a parameter. Question 2 is false and question 3 is true.

Question 1: A point estimate is the value of a statistic that estimates the value of a parameter.A point estimate is a single number that is used to estimate the value of an unknown parameter of a population, such as a population mean or proportion

Question 2: False

Mu (μ) is not used to estimate X. Mu represents the population mean, while X represents the sample mean. The sample mean, X, is used as an estimate of the population mean, μ.

Question 3: True

Beta (β) is indeed used to estimate the population proportion (p) when conducting hypothesis testing on a sample. Beta represents the probability of making a Type II error, which occurs when we fail to reject a null hypothesis that is actually false. By calculating the probability of a Type II error, we indirectly estimate the population proportion, p, under certain conditions and assumptions.

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The sample mean is the average of selected observations taken from population, whereas the population mean averages of whole population Sample mean: \(\bar{x}\) = ∑x / n

Population mean: μ =  ∑x / N

Therefore, the sample mean is average of a sample selected from population whereas population mean is average of whole population.

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a) The possible sequences of Y and N are YY ,YN ,NY ,NN. b) The probability for each sequence:

P(YY) = P(Y) * P(Y) = 0.58 * 0.58 = 0.3364

P(YN) = P(Y) * P(N) = 0.58 * 0.42 = 0.2436

P(NY) = P(N) * P(Y) = 0.42 * 0.58 = 0.2436

P(NN) = P(N) * P(N) = 0.42 * 0.42 = 0.1764

c) The probability that neither of the two randomly selected high school seniors has texted (NN) is given by P(NN) = 0.1764.d) P(exactly one has texted) = P(YN) + P(NY) = 0.2436 + 0.2436 = 0.4872e)The probability that both seniors have texted (YY) is given by P(YY) = 0.3364.

a. If we randomly sample two high school seniors of driving age and record Y if a senior has texted while driving and N if not, the possible sequences of Y and N are:

YY ,YN ,NY ,NN

b. Assuming each outcome is independent, we can calculate the probability for each sequence:

P(YY) = P(Y) * P(Y) = 0.58 * 0.58 = 0.3364

P(YN) = P(Y) * P(N) = 0.58 * 0.42 = 0.2436

P(NY) = P(N) * P(Y) = 0.42 * 0.58 = 0.2436

P(NN) = P(N) * P(N) = 0.42 * 0.42 = 0.1764

c. The probability that neither of the two randomly selected high school seniors has texted (NN) is given by P(NN) = 0.1764.

d. The probability that exactly one out of the two seniors has texted can occur in two ways: YN or NY. So, the probability is the sum of these two probabilities:

P(exactly one has texted) = P(YN) + P(NY) = 0.2436 + 0.2436 = 0.4872

e. The probability that both seniors have texted (YY) is given by P(YY) = 0.3364.

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Answer:

C) Both (1 , 11) and (-1, -5)

Step-by-step explanation:

y = 8x + 3

Plugin x = 1  and y = 11 in the equation

11 = 8*1 + 3

11 = 8 +3

11 = 11

So, (1 , 11)  is a solution.

Plugin x = -1  and y =-5 ,

-5 = 8*(-1) + 3

-5 = -8 + 3

-5 = -5

So, (-1 , -5) is also a solution

Answer:

y-y1/x-x1=0

y-5/x--1=0

y-5/x+1=0

cross multiplying

y-5=0(x+5)

equation=y-5

The order of T is 82.

Let n and e be the order and number of edges of T respectively.

Also, let a, b, c and d be the number of vertices of degree 2, 3, 4 and 5 respectively.

Then, 50+a+b+c+d = n

And by the Handshake Lemma,  the sum of the degrees of all the vertices of T is twice the number of edges, so we have

50+2a+3b+4c+5d = 2e

Since a=b=c=d then we have the following two equations,

50+4a = n

50+14a = 2e

But, T is a tree so e=n=1, then the two equations become,

50+4a = n ...............(1)

50+14a = 2(n-1) .............(2)

Solving for a in (1) we get,

a = \(\frac{n-50}{4}\)

Plugging a into (2) we get,

50+14(\(\frac{n-50}{4}\)) = 2(n-1)

50+3.5n-175 = 2n-2

3.5n-2n = 175-50-2

1.5n = 123

n = \(\frac{123}{1.5}\) = 82

Therefore, the order of T is 82.

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