its really confusing ​

The series converges absolutely for all x inside the interval of convergence, (-12, 10), and converges conditionally for x = -12 and x = 10. The radius of convergence is R = 11, and the interval of convergence is (-12, 10).

To determine the radius of convergence, we can use the ratio test. The ratio test states that a series ∑aₙ converges if the limit of the absolute value of the ratio of consecutive terms is less than 1:

lim (n→∞) |aₙ₊₁ / aₙ| < 1

If the limit is greater than 1, the series diverges, and if it is equal to 1, the test is inconclusive. Applying the ratio test to the given series, we have:

lim (n→∞) |((-1)^(n+2) x (x+1)ⁿ⁺¹ x n¹/ⁿ x 11ⁿ) / ((-1)ⁿ⁺¹ x (x+1)ⁿ x (n+1)¹/ⁿ⁺¹) x 11ⁿ⁺¹)|

= lim (n→∞) |(x+1) / 11| x (n / (n+1))¹/ⁿ⁺¹)

We can simplify this expression by taking the natural logarithm and using L'Hopital's rule:

ln lim (n→∞) |(x+1) / 11| x (n / (n+1))¹/ⁿ⁺¹)

= ln |(x+1) / 11| x lim (n→∞) (1/(n+1) - 1/n)/(1/(n+1) x ln(n/(n+1)))

= ln |(x+1) / 11| x lim (n→∞) (1 + ln(n/(n+1)))

= ln |(x+1) / 11|

So the series converges if |(x+1) / 11| < 1, which is equivalent to -12 < x < 10. Therefore, the radius of convergence is R = 11, and the interval of convergence is (-12, 10).

To determine the values of x for which the series converges absolutely, we need to consider the absolute value of the series:

∑n = 1 to infinity |((-1)ⁿ⁺¹ x (x+1)ⁿ) / (n¹/ⁿ x 11ⁿ)|

= ∑n = 1 to infinity [(x+1)/11]ⁿ / (n¹/ⁿ)

The term (x+1)/11 is less than 1 inside the interval of convergence, so we can apply the ratio test again:

lim (n→∞) |[(x+1)/11]ⁿ⁺¹ / [(n+1)¹/ⁿ⁺¹)] / [(x+1)/11]ⁿ / [n¹/ⁿ]|

= lim (n→∞) (x+1)/11 x (n/(n+1))¹/ⁿ⁺¹)

= (x+1)/11 < 1

Therefore, the series aₙ > aₙ₊₁ for all n, and lim (n→∞) aₙ = 0, so the series converges.

However, the alternating series test only guarantees convergence, not absolute convergence.

To determine whether the series converges absolutely or conditionally, we need to check whether the series ∑n = 1 to infinity |aₙ| converges.

We already showed that the series ∑n = 1 to infinity aₙ converges absolutely inside the interval of convergence, (-12, 10). Therefore, the series converges conditionally for all x in the boundary of the interval of convergence, x = -12 and x = 10.

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Answer:

$6.80

Step-by-step explanation:

1.79 x 3.8 = 6.802

So $6.80

Summary:

(i) To find the value of α such that the vectors v1, v2, and v3 are linearly dependent, we can set up a system of equations and solve for α.(ii) The Basis Theorem states that any set of linearly independent

(i) To check if v1, v2, and v3 are linearly dependent, we can set up the following equation:

c1v1 + c2v2 + c3v3 = 0,

where c1, c2, and c3 are constants. Substituting the given values of v1, v2, and v3, we have:

c1(3,7,7) + c2(1,4,4) + c3(α,1,1) = 0.

Simplifying this equation, we get the following system of equations:

3c1 + c2 + αc3 = 0,

7c1 + 4c2 + c3 = 0,

7c1 + 4c2 + c3 = 0.

We can solve this system of equations to find the value of α that satisfies the condition.

(ii) The Basis Theorem states that any set of linearly independent vectors that span a vector space can be used as a basis for that vector space. By applying the Basis Theorem to the vectors v1, v2, and v3, we can check if they form a basis for ℝ3. If they do, we can find the coordinates of a given vector, such as (2,3,5), in terms of the basis using Gaussian Elimination.

To apply Gaussian Elimination, we set up the augmented matrix [v1 | v2 | v3 | b], where b is the given vector (2,3,5). Then we perform row operations to obtain the row-echelon form of the augmented matrix. The resulting matrix will allow us to determine the coordinates of b in terms of the basis vectors.

By performing the Gaussian Elimination process, we can find the coordinates of (2,3,5) in terms of the basis vectors.

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There is no value of α that makes the vectors linearly dependent, and the basis for ℝ³ using the vectors [377, 148, 11α] is {v₁, v₂, v₃}, with the coordinates of [2, 3, 5] in terms of the basis found through Gaussian Elimination.

(i) To find the value of α such that vectors v₁, v₂, and v₃ are linearly dependent, we need to determine if there exist scalars a, b, and c, not all zero, such that a(v₁) + b(v₂) + c(v₃) = 0. Substituting the given values, we have a(377) + b(148) + c(11α) = 0. By solving this equation, we can find the value of α that satisfies the condition for linear dependence.

(ii) The Basis Theorem states that any set of linearly independent vectors that spans a vector space forms a basis for that vector space. Using a different value of α than the one found in (i), we can apply the Basis Theorem to determine a basis for ℝ³ using the vectors v₁, v₂, and v₃.

By performing Gaussian Elimination or row reduction on the augmented matrix [v₁ v₂ v₃], we can determine the basis vectors. The coordinates of vector [2 3 5] in terms of the basis can be found by solving the system of equations formed by equating the linear combination of the basis vectors to [2 3 5].

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Answer:

√625x12y8

Rewrite 625x12y8

as (5x3y2)4

.

4√(5x3y2)4

Pull terms out from under the radical, assuming positive real numbers.

5x3y2

Step-by-step explanation:

took the test :-) it's A. 5x^2|y^3|

Answer:

43

Step-by-step explanation:

Answer:

292 in² (3 s.f.)

Step-by-step explanation:

The figure is made up of 2 semicircles and a rectangle.

Please see attached picture for full solution.

Answer:

3*3*5*7

Step-by-step explanation:

hope it helps you out

please mark brainiest

Answer:

315 = 5*3*7*3

Step-by-step explanation:

The probability of event A and event B is 6.

Given that, P(A)=6, P(B)=20 and P(A∩B)=6.

P(A/B) Formula is given as, P(A/B) = P(A∩B) / P(B), where, P(A) is probability of event A happening, P(B) is the probability of event B.

P(A/B) = P(A∩B) / P(B) = 6/20 = 3/10

We know that, P(A and B)=P(A/B)×P(B)

= 3/10 × 20

= 3×2

= 6

Therefore, the probability of event A and event B is 6.

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Using the binomial distribution, it is found that there is a 0.8909 = 89.09% probability that both outcome Y and outcome Z occur at least once.

The trials are independent, hence the binomial distribution can be used.

The formula is:

\(P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}\)

\(C_{n,x} = \frac{n!}{x!(n-x)!}\)

The parameters are:

In this problem:

For the probability that outcome Y does not happen, we have that:

The probability is P(X = 5), that is, all five outcomes are either X or Z, then:

\(P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}\)

\(P(X = 5) = C_{5,5}.(0.6)^{5}.(0.4)^{0} = 0.0778\)

For the probability that outcome Z does not happen, we have that:

The probability is P(X = 5), that is, all five outcomes are either X or Y, then:

\(P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}\)

\(P(X = 5) = C_{5,5}.(0.5)^{5}.(0.5)^{0} = 0.0313\)

Then, the probability that at least one does not happen is:

\(p = 0.0778 + 0.0313 = 0.1091\)

The probability that both happen is:

\(1 - p = 1 - 0.1091 = 0.8909\)

0.8909 = 89.09% probability that both outcome Y and outcome Z occur at least once.

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Using implicit differentiation dz/dt = -34

Differentiation is the process of finding the derivative of a function.

Since x² + y² + z² = 9, dx/dt = 7, and dy/dt = 6, we desire to find dz/dt when (x, y, z) = (2, 2, 1).

Thus, we now differentiate implicitly and also we apply the chain rule, thus we have

x² + y² + z² = 9

d(x² + y² + z²)/dt = d9/dt

dx²/dx × dx/dt  + dy²/dy × dy/dt + dz²/dz × dz/dt = d9/dt

2xdx/dt  + 2ydy/dt + 2zdz/dt = 0

xdx/dt  + ydy/dt + zdz/dt = 0

we now make dz/dt subject of the formula, thus, we have

zdz/dt = -(xdx/dt  + ydy/dt)

dz/dt =  -(xdx/dt  + ydy/dt)/z

Given that

Substituting the values of the variables into the equation, we have that

dz/dt =  -(xdx/dt  + ydy/dt)/z

dz/dt =  -(2 × 7  + 2 × 6)/1

= -(12 + 14)/1

= - 26

So, dz/dt = -26

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The question is incomplete. Here is the complete question

If  x² + y² + z²  = 9, dx/dt = 7, and dy/dt = 6, find dz/dt when (x, y, z) = (2, 2, 1).

Answer:

Step-by-step explanation:

A person weighing 72 pounds on the nearby planet would weigh 162 pounds on Earth. Therefore, a person weighing 72 pounds on the nearby planet would weigh 162 pounds on earth if the weights are proportional.

If a person who weighs 198 pounds on earth would weigh 88 pounds on a nearby planet, then the ratio of their weight on earth to their weight on the nearby planet would be:

198/88 = 2.25

So, if we want to find out what a person weighing 72 pounds on the nearby planet would weigh on earth, we can set up a proportion:

198/88 = x/72

where x is the weight of the person on earth.

To solve for x, we can cross-multiply:

198 * 72 = 88 * x

14256 = 88x

x = 162

Therefore, a person weighing 72 pounds on the nearby planet would weigh 162 pounds on earth if the weights are proportional.

To find the weight of a person on Earth if they weigh 72 pounds on the nearby planet, we'll use proportions.

Let x be the weight of the person on Earth. We can set up the proportion as follows:

198 pounds (Earth) / 88 pounds (nearby planet) = x pounds (Earth) / 72 pounds (nearby planet)

To solve for x, cross-multiply:

198 * 72 = 88 * x

14256 = 88x

Now, divide both sides by 88 to find the weight on Earth:

x = 14256 / 88

x = 162

So, a person weighing 72 pounds on the nearby planet would weigh 162 pounds on Earth.

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Answer:

162 lb

Step-by-step explanation:

The weights are proportional, so set up a proportion and solve for the only unknown.

198 is to 88 as x is to 72

198/88 = x/72

99/44 = x/72

44x = 72 × 99

x = 7128/44

x = 162

Answer: 162 lb

Answer:

just divide i believe

162÷9

15÷5

84÷14

24÷12

To find the slope of the tangent line to a polar curve at a specific point, differentiate the equation with respect to theta and substitute the value of theta into the derivative. In this case, the slope is - square root(2).

To find the slope of the tangent line to the given polar curve at the point specified by the value of theta (r), you can use the following steps:
1. Determine the equation of the polar curve. This equation will provide the relationship between the radius (r) and the angle (theta).
2. Differentiate the equation with respect to theta. This will give you the derivative of r with respect to theta.
3. Plug in the value of theta at the specified point into the derivative obtained in step 2. This will give you the slope of the tangent line at that particular point.
The slope of the tangent line to the polar curve can vary depending on the specific equation of the curve and the value of theta. By following these steps, you will be able to find the slope of the tangent line at any given point on the curve.

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Answer:

-99.12525

Step-by-step explanation:

Answer:−  99.12525

Step-by-step explanation:

I multiplied it

Uisbg multiple line and angle theorems, the values of each of the 14 angles are given.

Using the geometry theorems :

1.)

∠2 is a right angle = 90°

2.)

∠1 + ∠2 + ∠3 = 180° (sum of angle on a straight line)

90 + ∠1 + 54° = 180°

∠1 = (180 - 144) = 36°

(∠2 and ∠5) ; (∠3 and ∠6) ; (∠1 and ∠4) are vertically opposite ;

Hence,

∠4 = 36°

∠5 = 90°

∠6 = 54°

∠3 = ∠8 = 54° (corresponding angles)

∠9 + ∠8 = 180° (sum of angle on a straight line)

∠9 = (180 - 54) =126°

∠7 = ∠9 = 126° (vertically opposite angles)

∠8 = ∠10 = 54° (vertically opposite angles)

∠4 = ∠13 = 36° (corresponding angles )

∠12 + ∠13 = 180° (sum of angle on a straight line)

∠12 = (180 - 36) =126°

∠11 = ∠13 = 36° (vertically opposite angles)

∠12 = ∠14 = 126° (vertically opposite angles)

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Answer:

33

Step-by-step explanation:

For the divisible by 4 part:

199 / 4 = 49.75, we just round that down to 49

For the divisible by 12 part:

199 / 12 equals around 16

So we get our answers and subtract them:

49 - 16 = 33

Therefore, 33 numbers less than 200 are divisible by 4 but not 12.

There are 33 numbers that are less than 200 and divisible 4 but not divisible 12.

Given that,
To determine how many natural numbers less than 200 are divisible by 4, but not divisible by 12.

The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.

Here,
Let the largest number divisible by 4 less than 200 is 198

The total numbers divisible by 4 and less than 200 is,
= 198 / 4 = 49

Similarly
The total number divisible by 12 and less than 200 is,
= 192/ 12 = 16
So the numbers that are less than 200 but divisible 4 not by 12 is,
= 49 - 16
= 33

Thus, there are 33 numbers that are less than 200 and divisible 4 but not divisible 12.

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Using the exponential growth formulas, we get the population at 10 years would be 2593.

What is exponential growth?

A process called exponential growth sees a rise in quantity over time. It happens when the derivative of a quantity's instantaneous rate of change with respect to time is equal to the original quantity.

Given Initial population is Pt= 1725.

The doubling time of the population is D=17

Now we need to find the population at time t=10

Using the formula,

\(P_{t}=P_{0} (2)^{\frac{t}{D} } \\P_{t}=1725 (2)^{\frac{10}{17} } \\P_{t}=1725 (2)^{0.588}\\P_{t}=1725 \space\ . \space\ 1.503\\P_{t}=2593\)

Hence the population of the island in 10  years would be 2593.

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Answer:

4pi/15=θ  - i think

Step-by-step explanation:

S = r*θ

r=9

s=12pi/5

12pi/5=9*θ

12pi/45=θ

4pi/15=θ

Answer:

1.96666666667 sorry I don't know the fraction

Step-by-step explanation:

1/3 + 5/6 + 4/3

Convert fractions to have common denominators

6/18 + 15/18 + 24/18 = 45/18

Convert to mixed number

2 9/18

Simplify

2 1/2

Answer:

3/8

Step-by-step explanation:

P(B) is given to you in the question.

In order to find the number of weeks w it will take Anita to reach her goal, we need to solve the following equation for w:

So, we can apply the same operations on both sides of the equation, until we isolate the variable w and find its value.

We obtain:

Therefore, Anita will have saved enough money for the snowboard in 5 weeks.

Answer:

15

----

153

15=6+3+1

153=18×17÷2

The integral 6 ∫ |3x - 3| dx can be interpreted as the area between the curve y = |3x - 3| and the x-axis, multiplied by 6.

The integral [\(\int\limits(5 + \sqrt{(49 - 2z^2)} )\) dz can be interpreted as the area between the curve \(y = 5 + \sqrt{(49 - 2z^2)}\) and the z-axis.

Now let's calculate the integrals in detail:

For the integral 6 ∫ |3x - 3| dx, we can split the integral into two parts based on the absolute value function:

6 ∫ |3x - 3| dx = 6 ∫ (3x - 3) dx for x ≤ 1 + 6 ∫ (3 - 3x) dx for x > 1

Simplifying each part, we have:

\(6 \int\limits (3x - 3) dx = 6 [x^2/2 - 3x] + C for x \leq 1\\6 \int\limits (3 - 3x) dx = 6 [3x - x^2/2] + C for x \geq 1\)

Combining the results, the final integral is:

\(6 \int\limits |3x - 3| dx = 6 [x^2/2 - 3x] for x \leq 1 + 6 [3x - x^2/2] for x > 1 + C\)

For the integral [ ∫ (5 + √(49 - 2z^2)) dz, we can simplify the square root expression and integrate as follows:

\([ \int\limits (5 + \sqrt{(49 - 2z^2)}dz = [5z + (1/3) * (49 - 2z^2)^{3/2}] + C\)

Therefore, the final result of the integral is:

\([ \int\limits (5 + \sqrt{(49 - 2z^2)}dz = [5z + (1/3) * (49 - 2z^2)^{3/2}] + C\)

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Pua will use 13, 1/4 cups of oatmeal.

The division in mathematics is one kind of operation. In this process, we split the expressions or numbers into the same number of parts.

Given:

Pua needs 3\(\frac{1}{4}\) cups of oatmeal.

The number of 1/4 cups of oatmeal,

she will use,

= 13/4 ÷ 1/4

= 13

Therefore, she will use 13 cups.

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The depth of the water in the cone-shaped tank is increasing at a rate of approximately 1.385 meters per second.

To determine the rate at which the depth of the water is changing, we can use related rates. Let's denote the depth of the water as h(t), where t represents time. We are given that dh/dt (the rate of change of h with respect to time) is 12 m/sec, and we want to find dh/dt when h = 18 meters.

To solve this problem, we can use the volume formula for a cone, which is V = (1/3)πr^2h, where r is the base radius and h is the depth of the water. We can differentiate this equation with respect to time t, keeping in mind that r is a constant (since the base radius does not change).

By differentiating the volume formula with respect to t, we get dV/dt = (1/3)πr^2(dh/dt). Now we can substitute the given values: dV/dt = 12 m/sec, r = 26 meters, and h = 18 meters.

Solving for dh/dt, we have (1/3)π(26^2) (dh/dt) = 12 m/sec. Rearranging this equation and solving for dh/dt, we find that dh/dt is approximately 1.385 meters per second. Therefore, the depth of the water in the tank is increasing at a rate of about 1.385 meters per second.

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If 700% of a number is 1540, then the number is 220

The given percentage = 700%

The percentage of a number is defined as the ratio that can be expressed as the fraction of 100.

Consider the number as x

Here the 700% of the number is 1540

Then the equation will become

x × 700% = 1540

Solve the equation and find the value of x

x × (700/100) = 1540

Divide the terms first

x × 7 = 1540

Move the 7 to the right hand side of the equation

x = 1540/7

Divide the terms

x = 220

Hence, if 700% of a number is 1540, then the number is 220

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The probability P(n < 10 | n is prime) is 4/6, which simplifies to 2/3 or approximately 0.67 (rounded to the nearest hundredth).

To find the probability P(n < 10 | n is prime), we need to determine the number of prime integers less than 10 and divide it by the total number of integers from 1 to 15 that are prime.

The prime numbers less than 10 are 2, 3, 5, and 7. So, there are 4 prime numbers less than 10.

The total number of integers from 1 to 15 that are prime is 6 (2, 3, 5, 7, 11, and 13).

As a result, the chance P(n 10 | n is prime) is 4/6, which can be expressed as 2/3 or, rounded to the nearest hundredth, as around 0.67.

Thus, 0.67 is the answer.

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