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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

Answer:

The building is taller than 12ft is b>12

The box contains more than 11 books is b>11

Answer:

b<11 is The board is shorter than 11 cm

b>11 is The box contains more than 11 books

b<12 is There are fewer than 12 beetles in a jar

b>12 is The building is taller than 12 ft.

Step-by-step explanation:

Therefore, the due date of the note is August 9 adding the number of days in the note's term to the note's date adding the number of days in the note's term to the note's date.

To determine the due date of the note, we need to add the number of days in the note's term to the note's date.

Given:

Note term: 120 days

Note date: April 9

To find the due date, we add 120 days to April 9.

April has 30 days, so we can calculate the due date as follows:

April 9 + 120 days = April 9 + 4 months = August 9

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Answer:

x = 7

Step-by-step explanation:

since the points all lie on the same line then the slopes of adjacent points will have the same slope.

calculate the slope using R and S then equate to slope using R and T or S and T

calculate slope using slope formula

m = \(\frac{y_{2}-y_{1} }{x_{2}-x_{1} }\)

with (x₁, y₁ ) = R (- 5, - 3 ) and (x₂, y₂ ) = S (- 1, - 1 )

\(m_{RS}\) = \(\frac{-1-(-3)}{-1-(-5)}\) = \(\frac{-1+3}{-1+5}\) = \(\frac{2}{4}\) = \(\frac{1}{2}\)

Repeat with (x₁, y₁ ) = R (- 5, - 3 ) and (x₂, y₂ ) = T (x, 3 )

\(m_{RT}\) = \(\frac{3-(-3)}{x-(-5)}\) = \(\frac{3+3}{x+5}\) = \(\frac{6}{x+5 }\)

equating \(m_{RS}\) and \(m_{RT}\)

\(\frac{6}{x+5}\) = \(\frac{1}{2}\) ( cross- multiply )

x + 5 = 12 ( subtract 5 from both sides )

x = 7

Answer:  3.2 yd

Step-by-step explanation:

Notice that TWV is a right triangle.  

Segment TU is not needed to answer this question.

∠V = 32°, opposite side (TW) is unknown, hypotenuse (TV) = 6

\(\sin \theta=\dfrac{opposite}{hypotenuse}\\\\\\\sin 32=\dfrac{\overline{TW}}{6}\\\\\\6\sin 32=\overline{TW}\\\\\\\large\boxed{3.2=\overline{TW}}\)

Answer:

D. 0

General Formulas and Concepts:

Algebra I

Pre-Calculus

Calculus

Derivatives

Derivative of sec(x) = sec(x)tan(x)

Step-by-step explanation:

Step 1: Define

\(\displaystyle h(x) = \sec x\)

Step 2: Differentiate

Step 3: Solve

∴ h'(0) is equal to 0.

___

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___

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

Answer:

9

Step-by-step explanation:

6+3=9

Answer:

9 is the answer and I don't know what is hsjjsjsjjw.

Step-by-step explanation:

Do the math! :)

Answer:

Slope: 2

Step-by-step explanation:

What is cross-section slicing ?

As per the figure, cross-sectional slicing means the division of the three-dimension into half and they are divided into the 3D view thus forming a

rectangular prisms.

All the 3 dimension figures include the cone, rectangle, and triangle. The volume of space that the cross-section of the plane makes tells us about the figure.

A slice parallel to the base of the rectangular prism, would result in two rectangular prisms. These two rectangular prisms would be congruent to each other as it is a perfect slice through the middle of dimension : 3cm x 4cm x 2cm.

A slice perpendicular to the base of the rectangular prism, would result in two rectangular prisms. These two rectangular prisms would be congruent to each other as it is a perfect slice through the middle of dimension : 1.5cm x 4cm x 2cm.

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The expression 148 + 19x³ represents the sum of 148 and the product of a number raised to the third power and 19.

To write the phrase "the sum of 148 and the product of a number raised to the third power and 19" in symbols using the variable x, we can write it as:

148 + 19x³

Here, we are adding 148 to the product of 19 and x raised to the third power. This expression represents the sum of 148 and the product of a number raised to the third power and 19. We can substitute any value for x to get the result of the expression. For example, if x is 2, then the expression becomes:

148 + 19(2³) = 148 + 19(8) = 300

Therefore, the sum of 148 and the product of a number raised to the third power and 19 is represented by the expression 148 + 19x³.

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Answer:

\(f(x)=-2(x+2)(x-4)\)

Step-by-step explanation:

In factored form, \(f(x)=a(x+2)(x-4)\).

To solve for \(a\), use the point \((-1,10)\).

\(10=a(-1+2)(-1-4) \implies a=-2 \\ \\ \therefore f(x)=-2(x+2)(x-4)\)

Answer:

- 12(x + 2)

Step-by-step explanation:

- 12x - 24 ← factor out the common factor of - 12 from each term

= - 12(x + 2)

(a) The probability of winning one game is 64%. To win four games in a row, we need to calculate (0.64)^4, which equals 0.167, or 16.7%. Therefore, the probability that the Dodgers will win four games in a row is 16.7%.

(b) Following the same logic, to win seven games in a row, we need to calculate (0.64)^7, which equals 0.059, or 5.9%. Therefore, the probability that the Dodgers will win seven games in a row is 5.9%.

(e) To calculate the probability of losing at least one game out of seven, we need to calculate the probability of winning all seven games, and then subtract that from 1. The probability of winning all seven games is (0.64)^7, which we calculated earlier as 5.9%. Subtracting that from 1, we get 0.941, or 94.1%. Therefore, the probability that the Dodgers will lose at least one of their next seven games is 94.1%.

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Answer:

4

Step-by-step explanation:

the above question simply means that

{X+3X-K X+4} - k

is exactly divisible by (X -2)

to find the value of K, put X= 2

2+3(2)-K(2)+4-k=0

2+6-2K+4-K = 0

12-3K = 0

12 = 3K

K = 12/3 = 4

K= 4

Answer:

The remainder when the polynomial f(x)=2x3+px2+qx+18 is divided by (x-1) is 10, when it is divided by (x+1) the remainder is R find (1) The values of p and q (2) The zeros of f(x)

Step-by-step explanation:

The probability that the sample mean is 36 or more pieces of candy is approximately 18.67%.

To calculate the probability that the sample mean is 36 or more pieces of candy, we need to use the properties of the normal distribution. Given that the population mean is 35 pieces of candy and the standard deviation is 5 pieces, we can use the Central Limit Theorem.
The Central Limit Theorem states that as the sample size increases, the distribution of the sample means approaches a normal distribution, regardless of the shape of the population distribution. Since the sample size is 40, we can assume a normal distribution.
To calculate the probability, we need to standardize the sample mean using the formula z = (x - μ) / (σ / √n),

where x is the sample mean, μ is the population mean, σ is the standard deviation, and n is the sample size.
In this case, x = 36, μ = 35,

σ = 5, and n = 40.

Plugging these values into the formula, we get z = (36 - 35) / (5 / √40)

≈ 0.8944.
To find the probability that the sample mean is 36 or more, we need to calculate the area under the normal curve to the right of z = 0.8944. Using a standard normal table or a calculator, we find that the probability is approximately 0.1867, or 18.67%.
Therefore, the probability that the sample mean is 36 or more pieces of candy is approximately 18.67%.

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Answer:

A

Step-by-step explanation:

Answer:

y = 1/3x + 3

Step-by-step explanation:

Slope form of a equation: y = mx + b. The slope is 1/3, so y = 1/3x + b

Solve for b by plugging the (x, y) coordinate into the equation.

y = 1/3x + b

2 = 1/3(-3) + b

2 = -3/3 + b

2 = -1 + b

b = 3

So your answer is y = 1/3x + 3

Answer:

  (c)  f(g(3)) > g(f(3))

Step-by-step explanation:

The relationship between the function values can be found by evaluating the functions.

The order of operations tells us we must first evaluate g(3).

  g(x) = -3x +15

  g(3) = -3(3) +15 = 6

Then we can evaluate f(x) for x=6:

  f(x) = 7x

  f(6) = 7(6) = 42

So, the composition is ...

  f(g(3)) = 42

As above, we must first evaluate f(3):

  f(3) = 7(3) = 21

Then we can evaluate g(x) for x=21:

  g(21) = -3(21) +15 = -63 +15 = -48

That means the composition is ...

  g(f(3)) = -48

The first is greater than the second.

  42 > -48

  f(g(3)) > g(f(3))

The set of points in space whose coordinates satisfy a given combination of equations and inequalities can be described by a geometric shape or region in 3-dimensional space.

Enumerate the collection of spatial points whose coordinates fulfil the specified set of equations and inequalities.

The equations and inequalities will define the properties of the region, such as its shape, size, and location.

For example, if we have the equation x + y + z = 3 and the inequality x^2 + y^2 <= 1, the set of points that satisfy these conditions would be the points on or inside a sphere of radius 1 centered at the point (0,0,-1) in the x-y plane, and points on the plane x+y+z = 3.

Another example, if we have the equation x + y = 4 and the inequality y >= 2x-5, the set of points that satisfy these conditions would be the points on the line x+y = 4, above or on the line y = 2x-5.

It's important to note that the type of shape or region that is formed by the equations and inequalities will depend on the specific equations and inequalities being used and the number of variables in the problem.

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Answer:

\(\frac{a}{6}\) = \(\frac{15}{2}\)

2a = 6 · 15

2a = 90

a = 45

Step-by-step explanation:

Answer:

First, you need to calculate your grade in percentages. The total answers count 70 - it's 100%, so we to get a 1% value, divide 70 by 100 to get 0.70. Next, calculate the percentage of 15: divide 15 by 1% value (0.70), and you get 21.43% - it's your percentage grade.

Step-by-step explanation:

The sample's standard deviation is equal to 12. Thus, choice B is the right choice.

The variance of a sample of 144 observations equals 576.

Variance S² = 144

The standard deviation is a measure of how spread out a set of data is. It is calculated by taking the square root of the variance, which is the average of the squared differences from the mean.

The standard deviation can be used to measure the amount of variation or dispersion of a set of data values from the mean.

Standard Deviation = √Variance

Standard Deviation = √144

Standard Deviation = 12

The standard deviation of the sample equals 12. So the option B is correct.

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The complete question is:

The variance of a sample of 144 observations equals 576. The standard deviation of the sample equals:

A. 441

B. 12

C. 63,504

D. 21

(a) Player Column's best response is given by:

BR_Column(p) = { L if p < 1/2, R if p > 1/2 (indifferent if p = 1/2)

(b) Where both players are indifferent between their available strategies.

(c)  The normal form representation of the game is above.

(d) No player can gain an advantage by deviating from this strategy.

This equilibrium results in an expected payoff of 0 for each player.

(a) To derive the best response functions, we need to find the strategies that maximize the payoffs for each player given the mixed strategy of the other player.

Player Row's best response function:

If Player Column chooses L with probability q, Player Row's expected payoff for choosing U is 0q + 2(1-q) = 2 - 2q.

If Player Column chooses R with probability 1-q, Player Row's expected payoff for choosing U is 0*(1-q) + 1*q = q.

Therefore, Player Row's best response is given by:

BR_Row(q) = { U if q < 1/3, D if q > 1/3 (indifferent if q = 1/3)

Player Column's best response function:

If Player Row chooses U with probability p, Player Column's expected payoff for choosing L is 0p + 2(1-p) = 2 - 2p.

If Player Row chooses D with probability 1-p, Player Column's expected payoff for choosing L is 0*(1-p) + (-1)*p = -p.

Therefore, Player Column's best response is given by:

BR_Column(p) = { L if p < 1/2, R if p > 1/2 (indifferent if p = 1/2)

Plotting the best response functions on a graph with p on the horizontal axis and q on the vertical axis will result in two line segments: BR_Row(q) is horizontal at U for q < 1/3 and horizontal at D for q > 1/3, while BR_Column(p) is vertical at L for p < 1/2 and vertical at R for p > 1/2.

The two segments intersect at the point (p, q) = (1/2, 1/3).

(b) To find the Nash equilibria, we look for the points where the best response functions intersect. In this case, the only Nash equilibrium is at (p, q) = (1/2, 1/3), where both players are indifferent between their available strategies.

Now let's move on to the rock-paper-scissors-lizard game:

(c) The normal form representation of the game can be written as follows:

    R    P    S    L

------------------------

R | 0,0 -1,1 1,-1 1,-1

P | 1,-1 0,0 -1,1 1,-1

S | -1,1 1,-1 0,0 -1,1

L | -1,1 -1,1 1,-1 0,0

(d) To find the Nash equilibria, we look for any strategy profiles where no player can unilaterally deviate to improve their payoff.

In this game, there are no pure strategy Nash equilibria since each strategy can be countered by another strategy with a higher payoff.

However, there is a mixed strategy Nash equilibrium where each player chooses their actions with equal probabilities: r = p = s = l = 1/4.

In this case, no player can gain an advantage by deviating from this strategy.

This equilibrium results in an expected payoff of 0 for each player.

In summary, the rock-paper-scissors-lizard game has a unique mixed strategy Nash equilibrium where each player randomly chooses their actions with equal probabilities.

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According to the given information, we can draw the following

As you can observe in the image, MP is formed by OM plus PO by the sum of segments theorem, so using the given information, we have

Answer:

36.98 inches each

Step-by-step explanation:

He uses 857.77 inches of fabric on headbands for 29 players and 2 coaches

29 players + 2 couches = 31 people

Inches per person = 857.77 inches / 31 people

= 27.67 inches for headbands

He also uses 269.99 inches of fabric on wristbands for just the players.

Inches per person = 269.99 inches / 29 players

= 9.31 inches for wristband

How much fabric was used on a headband and wristband for each player = 27.67 inches for headbands + 9.31 inches for wristband

= 36.98 inches each

Answer:

6 ballpoints pens is the better buy

Step-by-step explanation:

$4.80 x 6 = $28.80

$6.48 x 8 = $51.84

It's cheaper

Answer:

6

Step-by-step explanation:

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