Find all values of a and b (if any) so that the given vectors form an orthogonal set. (If an answer does not exist, enter DNE.) u_1 = [2 1 -1], u_2 = [4 -5 3], u_3 = [2 a b]

Answer:

$0.70

Step-by-step explanation:

Let the cost of an orange be represented by o and the cost of a mango be represented by m.

According to the problem, we can set up the following system of equations:

2o + 5m = 4.5

4o + 3m = 4.1

We can solve for o and m using elimination or substitution. Here's one way to solve it using elimination:

Multiply the first equation by 4 and the second equation by -2 to eliminate o:

8o + 20m = 18

-8o - 6m = -8.2

14m = 9.8

m = 0.7

Substitute m = 0.7 into one of the equations to solve for o:

2o + 5(0.7) = 4.5

2o + 3.5 = 4.5

2o = 1

o = 0.5

Therefore, the cost of an orange is $0.50 and the cost of a mango is $0.70.

Hope this helps!

The closer the value of r to O the greater the variation around the line of best fit. Different... Are there guidelines to interpreting Pearson's correlation coefficient? Yes, the following guidelines have been proposed: ...

If the first equation is 3x + 2y = 5, a second equation that would make this system have no solution is: A. 6x + 4y = 10.

If the first equation is y = 2x - 5, a second equation that would make this system have no solution is: A. 4x - 2y = 10.

The statement that correctly describes the solution to this system of equations is: A. there is no solution.

The statement that correctly describes the solution to this system of equations is: A. there is no solution.

The statement that correctly interprets Xaysha's solution is: C. there is no solution since 5 = 7 is a false statement.

The statement that correctly interprets Sophina's solution is: A. the solution is x = 0.

In Mathematics, no solution is sometimes referred to as zero solution, and an equation is said to have no solution when the left hand side and right hand side of the equation are not the same or equal.

This ultimately implies that, a system of equations would have no solution when the line representing each of the equations are parallel lines and have the same slope or coincide i.e both sides of the equal sign are the same and the variables cancel out.

For Sophina's solution, we have;

7x = 6x

7x - 6x = 0

x(7 - 6) = 0

x = 0

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It is a rectangle because the opposite sides were parallel and congruent

Given data,

Let the quadrilateral be represented as WXYZ

Now , the line WX is parallel and congruent to the side YZ

So, they have the same slope

And , the line segment WZ is parallel and congruent to the side XY

So , they have the same slope

Therefore , the quadrilateral is a rectangle

Hence , the figure is a rectangle and the opposite sides have same slope

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The probability that Lin gets another turn is 1/60

Here, Lin will get a turn if both the cube and card have the same number.

A cube is numbered as 1, 2, 3, 4, 5, and 6

And the deck of 10 cards numbered 1 through 10

We can observe that there are only 6 (1 to 6) numbers which can follow above condition.

So, the chances of getting 6 in dice equals to 1/6

and the chances of getting 6 in card equals 1/10

Thus, the chance of getting both at once would be,

1/6 × 1/10

= 1/60

Therefore, the required probability is 1/60

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An expression to find the distance between the two jellyfish is B - A.

The distance is 2.0 meters.

In Mathematics and Geometry, a number line simply refers to a type of graph with a graduated straight line which comprises both positive and negative numbers that are placed at equal intervals along its length.

This ultimately implies that, a number line primarily increases in numerical value towards the right from zero (0) and decreases in numerical value towards the left from zero (0).

Therefore, the required expression for the distance between the two jellyfish is given by;

Distance = B - A

Distance = 0.5 - (-1.5)

Distance = 0.5 + 1.5

Distance = 2.0 m.

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The region in the plane consists of all points with polar coordinates (r,θ) such that 0 ≤ r < 7 and π ≤ θ ≤ 3π/2 is the shaded region in the fourth quadrant bounded by the circle with radius 7 and the positive x-axis extended to the origin.

In polar coordinates, a point in the plane is represented by its distance from the origin (r) and the angle it makes with the positive x-axis (θ). The given conditions are 0 ≤ r < 7 and π ≤ θ ≤ 3π/2.

The condition 0 ≤ r < 7 means that the points must be inside the circle of radius 7 centered at the origin. The condition π ≤ θ ≤ 3π/2 means that the points must be in the fourth quadrant and lie between the angles π and 3π/2 measured from the positive x-axis.

Therefore, the shaded region in the fourth quadrant bounded by the circle with radius 7 and the positive x-axis extended to the origin is the region in the plane consisting of all points with polar coordinates (r,θ) such that 0 ≤ r < 7 and π ≤ θ ≤ 3π/2.

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The function f(x, y, z) that satisfies F = ∇f is f(x, y, z) = xyz + 3xz + 3yz² + 6zk + C, where C is a constant. The integral of ∇f · dr along C is given by:

∫∇f · dr = f(x2, y2, z2) - f(x1, y1, z1) = f(5, 6, 3) - f(3, 0, -1).

To find the function f(x, y, z) such that F = ∇f, we need to determine the gradient of f and equate it to F.

Gradient of f = ∇f = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k

Comparing the components of ∇f with F:

∂f/∂x = yz (1)

∂f/∂y = xz (2)

∂f/∂z = xy + 6z (3)

To solve for f, we integrate each of the partial derivatives with respect to their respective variables.

From equation (1), integrating with respect to x:

f(x, y, z) = xyz + g(y, z), where g(y, z) is a function of y and z.

Taking the partial derivative of f(x, y, z) with respect to y and comparing with equation (2):

∂f/∂y = xz + (∂g/∂y) = xz

∂g/∂y = 0

Integrating g(y, z) with respect to y:

g(y, z) = yz² + h(z), where h(z) is a function of z.

Taking the partial derivative of f(x, y, z) with respect to z and comparing with equation (3):

∂f/∂z = xy + 6z + (∂h/∂z) = xy + 6z

∂h/∂z = 0

Integrating h(z) with respect to z:

h(z) = 6zk + C, where C is a constant.

Substituting the expressions for g(y, z) and h(z) back into f(x, y, z):

f(x, y, z) = xyz + 3xz + 3yz² + 6zk + C

Therefore, the function f(x, y, z) that satisfies F = ∇f is f(x, y, z) = xyz + 3xz + 3yz² + 6zk + C, where C is a constant.

To evaluate the integral of ∇f · dr along the given curve C, we substitute the coordinates of the two endpoints of C into f(x, y, z) and calculate the difference.

Coordinates of the starting point of C: (x1, y1, z1) = (3, 0, -1)

Coordinates of the ending point of C: (x2, y2, z2) = (5, 6, 3)

Substituting the coordinates into f(x, y, z):

f(x1, y1, z1) = f(3, 0, -1)

f(x2, y2, z2) = f(5, 6, 3)

The integral of ∇f · dr along C is given by:

∫∇f · dr = f(x2, y2, z2) - f(x1, y1, z1) = f(5, 6, 3) - f(3, 0, -1)

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The required answer is = (n/(n+1))theta

theta = (n/(n+1))theta the sample maximum is an unbiased estimator of theta.

One unbiased estimator of theta is the sample maximum, which is defined as max(x1,...,xn). To show that this estimator is unbiased, we need to calculate its expected value and show that it equals theta.

Since x1,...,xn are i.i.d. random variables from the uniform distribution on the interval [0,theta], their probability density function is f(x) = 1/theta for 0 <= x <= theta and 0 otherwise.

The probability that the sample maximum is less than or equal to a given value x is the probability that all n samples are less than or equal to x. Since the samples are i.i.d., this probability is (x/theta)^n.

Therefore, the cumulative distribution function of the sample maximum is F(x) = (x/theta)^n for 0 <= x <= theta and 0 otherwise.

The probability density function of the sample maximum is the derivative of its cumulative distribution function, which is f(x) = (n/theta)(x/theta)^(n-1) for 0 <= x <= theta and 0 otherwise.

The expected value of the sample maximum is the integral of x times its probability density function from 0 to theta, which is

E[max(x1,...,xn)] = integral from 0 to theta of x*(n/theta)(x/theta)^(n-1) dx

= (n/theta) integral from 0 to theta of x^n/theta^n dx

= (n/theta) * [x^(n+1)/(n+1)theta^n)]_0^theta

= (n/(n+1))theta
Therefore, the sample maximum is an unbiased estimator of theta.

To find an unbiased estimator of theta, given that x1, ..., xn are i.i.d. random variables from the uniform distribution on the interval [0, theta], follow these steps:

1. Determine the sample maximum: Since the data is from a uniform distribution, the sample maximum (M) can be used as a starting point. M = max(x1, ..., xn).

2. Calculate the expected value of M: The expected value of the sample maximum, E(M), is given by the formula E(M) = n * theta / (n + 1).

3. Find an unbiased estimator: To find an unbiased estimator of theta, we need to adjust the expected value of M so that it equals theta. We can do this by solving for theta in the equation E(M) = theta:

theta = (n + 1) * E(M) / n

4. Replace E(M) with the sample maximum M: Since we are using the sample maximum as our estimator, we can replace E(M) with M in the equation:

theta_hat = (n + 1) * M / n

The unbiased estimator of theta is theta h at = (n + 1) * M / n, where M is the sample maximum and n is the number of i.d. random variables.

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The expected value of the function 3x + 2y, given a valid joint discrete distribution, can be calculated using the properties of expected values.

The expected value of a function of two discrete random variables x and y, denoted as E(3x + 2y), is the sum of the products of the possible values of x and y, weighted by their respective probabilities, in accordance with the joint distribution. Mathematically, it can be expressed as:

E(3x + 2y) = Σ[ (3x + 2y) × P(x, y) ]

where Σ denotes the sum over all possible values of x and y, and P(x, y) represents the joint probability of x and y.

To calculate the expected value, follow these steps:

Identify the possible values of x and y from the given joint discrete distribution.

Compute the joint probability P(x, y) for each combination of x and y using the provided distribution.

Multiply each value of x by 3, and each value of y by 2.

Multiply the result of step 3 by the corresponding joint probability P(x, y) from step 2.

Sum up all the products obtained in step 4 to get the final expected value.

Therefore, the expected value E(3x + 2y) can be found by performing the above steps and summing the resulting products, in accordance with the provided joint discrete distribution.

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Jenny paid $176 for 22 T-shirts, which is $8 per T-shirt .

The unitary method, commonly referred to as the unit rate or the single quantity method, is a mathematical approach for resolving issues requiring proportional connections between numbers. Finding the value of one unit of a quantity, which is frequently used as a reference or a benchmark, and utilising that value to compute or compare other numbers are both involved in this process.

In other words, you may compute the value or rate of one unit of a quantity using the unitary technique, and then use that rate to derive the value or rate of another quantity. This approach is frequently employed in a variety of real-world contexts, including the computation of costs, rates, ratios, and proportions.

Given:

Jenny paid $176 for 22 T-shirts.

To find the cost per T-shirt, we need to divide the total cost by the number of T-shirts.

Using Unitary method;

$176 ÷ 22 = $8 per T-shirt.

Therefore, D is correct statement: Jenny paid $176 for 22 T-shirts, which is $8 per T-shirt .

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Correct Question:

Jenny is in charge of ordering T-shirts for the math club at her school. If she paid $176 for 22 T-shirts, which of the following statements is true?

A. Jenny paid $176 for 22 T-shirts, which is $20 per T-shirt.

B. Jenny paid $176 for 22 T-shirts, which is $11 per T-shirt.

C. Jenny paid $176 for 22 T-shirts, which is $12 per T-shirt.

D. Jenny paid$176 for 22 T-shirts, which is $8 per T-shirt.

The determinant of the linear transformation t(f)=2f 3f' from p2 to p2 is 36.

To find the determinant of the linear transformation t(f)=2f 3f' from p2 to p2, we first need to represent the transformation as a matrix.

Let's start by choosing a basis for p2, say {1,x,x²}. Then, the linear transformation t can be represented by the matrix

[2 0 0]
[0 3 0]
[0 0 6]

To find the determinant of this matrix (and hence the determinant of the linear transformation), we can use the formula for the determinant of a 3x3 matrix:

det(A) = a11(a22a33 - a23a32) - a12(a21a33 - a23a31) + a13(a21a32 - a22a31)

Plugging in the entries of our matrix, we get:

det(t) = 2(3×6 - 0×0) - 0(2×6 - 0×0) + 0(2×0 - 3×0)
      = 36

Therefore, the determinant of the linear transformation t(f)=2f 3f' from p2 to p2 is 36.

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The percentage of middle schoolers is given as 45.1%

A percentage is a fractional expression, written as parts of every one-hundred, represented by the sign "%".

For instance, assume there are 20 red balls in a bag that comprises of 100 balls – providing us an opportunity to ascertain that the portion of red balls available in the aforementioned sack is twenty percent.

If you wish to calculate a percentage of numbers, then you must first divide the part by the entirety and multiply the result with one hundred. As an example, if we want to calculate what amount of hundred is constituted by twenty, then we simply divide twenty by hundred which gives us 0.2, and then further imply multiplication of it by 100 which equals twenty percent.

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ωc = 1/RC. where R is the resistance of the circuit and C is the capacitance. Once we know ωc, we can substitute it into the equation for vo(t) and evaluate the expression.

When ω=ωc, the output voltage can be found by substituting ωc for ω in the equation for vo(t):

vo(t) = a*cos(ωc*t + ϕ)v

The values of a, ω, and ϕ are given in the problem:

a > 0
-180∘ < ϕ ≤ 180∘
ω = angular frequency of the input signal

So, to find the output voltage when ω=ωc, we need to know the value of ωc. This value depends on the circuit parameters and can be calculated using the formula:

ωc = 1/RC

Without more information about the circuit, we cannot provide a specific answer for the output voltage when ω=ωc.

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The statement is expressed as:

[tex]y \alpha 1/x[/tex]

To convert to an equation introduce k, the constant of variation.

[tex]y=k * 1/x\\[/tex]

To find k use the condition that [tex]x = 12[/tex] when [tex]y = 5[/tex]

[tex]y=k/x[/tex]

[tex]5=k/12[/tex]

[tex]k=5*12[/tex]

[tex]k=60\\[/tex]

Therefore,  [tex]y =60/x[/tex] is the function.

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The answer is A. True. When working with big data, the sample size is so large that the variability in the data almost completely disappears.

Variability is the degree of difference between values in a set of data. It is a measure of how spread out the values are from the mean or average of the set.

When dealing with smaller datasets, there is typically more variability. This is because a smaller sample size does not represent the entire population of data, and therefore does not provide an accurate representation of the underlying population of data. This variability can make it more difficult to draw valid conclusions from the data.

However, when working with a large sample size, the variability virtually disappears. This is because the data is spread across so many data points that the differences between individual data points become negligible. The result is that the data becomes more consistent and easier to analyze.

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The power series expansion of the function f(x) = sinx. The correct answer is (b) 1-x^2/2 +x^4/24 -x^6/720.

To obtain this answer, we can use the power series expansion formula for sinx, which is given by

sinx = x - x^3/3! + x^5/5! - x^7/7! + ... .

Evaluating the first four terms of this expansion around x=0, we get

sinx = x - x^3/3! + x^5/5! - x^7/7! + ...

= x - (x^3/6) + (x^5/120) - (x^7/5040) + ...

= 1-x^2/2 +x^4/24 -x^6/720 + ...,

which is equivalent to option (b).

Therefore, the first four nonzero terms in the power series expansion of f(x) = sinx about x=0 are 1-x^2/2 +x^4/24 -x^6/720. The correct option is B).

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Answer:

7/24

Step-by-step explanation:

A = 7/8 × 2/3 × 1/2 = 14/48 = 7/24

The 96% confidence interval for the soda cans' diameters manufactured by CO 4 is approximately 51.1 mm to 52.9 mm.

To calculate the 96% confidence interval for the soda cans' diameters, we need to consider the sample mean, standard deviation, and sample size, as well as the appropriate Z-score for the desired level of confidence.

The terms you've provided are:
- Company (CO 4) - A company that manufactures soda cans.
- Diameter of 52 mm - The average diameter of the soda cans.
- Sample size of 18 - The number of soda cans in the sample.
- Standard deviation of 2.3 mm - The measure of dispersion in the sample.

Given the information, we first need to calculate the standard error (SE), which is the standard deviation (2.3 mm) divided by the square root of the sample size (18). This can be calculated as follows:

SE = 2.3 / √18 ≈ 0.54

For a 96% confidence interval, we use a Z-score of 2.05, which means we are 96% confident that the true population means lies within this interval. Now, we can calculate the confidence interval:

Lower limit = Sample mean - (Z-score × SE) = 52 - (2.05 × 0.54) ≈ 51.1 mm
Upper limit = Sample mean + (Z-score × SE) = 52 + (2.05 × 0.54) ≈ 52.9 mm

So, the 96% confidence interval for the soda cans' diameters manufactured by CO 4 is approximately 51.1 mm to 52.9 mm. This means we are 96% confident that the true average diameter of the soda cans produced by the company lies within this range.

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The maximum likelihood estimate of θ is 0.8.

The likelihood function for the sample is:

L(θ) = fY(2.3;θ) x fY(1.9;θ)  x fY(4.6;θ)

      = (2.3 e^(-2.3/theta) / (6 θ theta)) (1.9e^(-1.9/theta) / (6 theta)) (4.6 e^(-4.6/theta) / (6 theta))

Taking the natural logarithm of both sides, we get:

ln(L(θ)) = ln(2.3) - (2.3/θ) - 3ln(θ) - ln(6) + ln(1.9) - (1.9/θ) - 3ln(θ) - ln(6) + ln(4.6) - (4.6/θ) - 3ln(θ) - ln(6) = 3ln(2.3) - (2.3/θ) - 3ln(θ) - ln(6) + 3ln(1.9) - (1.9/θ) - 3ln(θ) - ln(6) + 3ln(4.6) - (4.6/θ) - 3ln(θ) - ln(6)

To find the maximum likelihood estimate of theta, we need to maximize this expression with respect to theta. We can do this by taking the derivative of the expression with respect to theta, setting it to zero, and solving for theta:

d/dθ ln(L(θ)) = (2.3/θ) - (3/θ) + (1.9/θ) - (3/θ) + (4.6/θ) - (3/θ) = 0 Simplifying this expression, we get:

= (2.3/θ) + (1.9/θ) + (4.6/θ)

= (9/theta)

Multiplying both sides by θ, we get:

2.3 + 1.9 + 4.6 = 9θ

Solving for θ , we get:

θ= (2.3 + 1.9 + 4.6) / 9 = 0.8

Therefore, the maximum likelihood estimate of θ is 0.8.

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Answer:

yes

Step-by-step explanation:

because it has a dramatic decrease in value

Sum of the series [infinity] ln n² / (8 + 7n²)² is π² / 6.

Series [infinity] 1/n² convergent to π² / 6.

We can use the limit comparison test.

First, note that both ln n² and (8 + 7n²)² are positive for all n ≥ 1.

Let a_n = ln n² / (8 + 7n²)².

Then, consider the series b_n = 1/n².

We know that b_n is a convergent p-series with p = 2.

Next, we take the limit of the ratio of a_n and b_n as n approaches infinity:

lim (n→∞) a_n / b_n = lim (n→∞) (ln n² / (8 + 7n²)²) / (1/n²)

Using L'Hôpital's rule twice, we can simplify this limit as follows:

= lim (n→∞) [(2/n) / (-28n / (8 + 7n²))]

= lim (n→∞) -14n / (8 + 7n²)

Since the numerator and denominator both approach infinity as n approaches infinity, we can apply L'Hôpital's rule again:

= lim (n→∞) -14 / (14n)

= 0

Since the limit is finite and positive, we can conclude that the series [infinity] ln n² / (8 + 7n²)² converges by the limit comparison test.

To find its sum, we can use a known result that the series [infinity] 1/n² converges to π² / 6.

Therefore, the sum of the series [infinity] ln n² / (8 + 7n²)² is π² / 6.

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The statement that is TRUE regarding boxes randomly sampled from the filling process is that in a very large sample of boxes, you are certain to get at least one underfilled box. This can be answered by the concept of Probability.

The probability of a box being underfilled is 0.10 or 10%. In a large sample of boxes, as the number of sampled boxes increases, the likelihood of encountering at least one underfilled box also increases. This is because the probability of at least one box being underfilled becomes virtually certain in a large sample size. As the sample size approaches infinity, the probability of encountering at least one underfilled box approaches 100%.

Therefore, in a very large sample of boxes, you are certain to get at least one underfilled box

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Answer:

16.67%

Step-by-step explanation:

[tex]\frac{40}{240}[/tex] × 100 = 16.67%

The given sequence an = (7n+2)/(8n) converges, and its limit is 7/8

To determine whether the sequence converges or diverges, and to find the limit if it converges, we will analyze the given sequence an = (7n + 2) / (8n).

Step 1: Simplify the sequence by dividing the numerator and the denominator by the highest power of n, which in this case is n^1.

an = (7 + 2/n) / (8)

Step 2: Take the limit of the sequence as n approaches infinity.

lim (n -> ∞) [(7 + 2/n) / 8]

Step 3: As n approaches infinity, the term 2/n approaches 0.

lim (n -> ∞) [(7 + 0) / 8]

Step 4: Simplify the limit.

lim (n -> ∞) [7 / 8] = 7/8

Since the limit exists and is a finite value (7/8), we can conclude that the sequence converges, and its limit is 7/8.

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To demonstrate that the limit lim(x,y)→(0,0) [tex](x^2)/(x^2y^2 + (x-y)^2)[/tex] does not exist, we can use two different paths:

Path 1: Let y = x In this path, we substitute y with x in the expression: lim(x,x)→ [tex](0,0) (x^2)/(x^2x^2 + (x-x)^2)[/tex] = lim(x,x)→ [tex](0,0) (x^2)/(x^4)[/tex]As x approaches 0, the expression simplifies to: lim(x→0) [tex](x^2)/(x^4)[/tex] = lim(x→0) [tex]1/x^2[/tex] When x approaches 0, [tex]1/x^2[/tex] goes to infinity.

Therefore, the limit along this path does not exist.

Path 2: Let y = 0 In this path, we substitute y with 0 in the expression: lim(x,0)→(0,0)[tex](x^2)/(x^2(0)^2 + (x-0)^2)[/tex] = lim(x,0)→ [tex](0,0) (x^2)/(x^2)[/tex] As x approaches 0, the expression simplifies to: lim(x→0) (x^2)/(x^2) = lim(x→0) 1

When x approaches 0, the expression equals 1, which is a finite value.

Since the limits along Path 1 and Path 2 are not equal, we can conclude that the limit lim(x,y)→(0,0) [tex](x^2)/(x^2y^2 + (x-y)^2)[/tex] does not exist.

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(1) To show that X has a subsequence X' such that either X' is in A or X' is in B, we can use the fact that A and B are subsets of the metric space (M,d) to construct two subsequences, one consisting of terms from A and the other consisting of terms from B.

Let X_A be the subsequence of X that consists of all terms in A, and let X_B be the subsequence of X that consists of all terms in B. If either of these subsequences is infinite, then we are done. Otherwise, both A and B are finite sets, and we can construct a subsequence X' by interleaving the terms from X_A and X_B in any way we choose.

For example, suppose A = {a1, a2, a3} and B = {b1, b2}. Then X_A = (a1, a2, a3) and X_B = (b1, b2), and we can construct the subsequence X' = (a1, b1, a2, b2, a3). This subsequence has terms from both A and B, but we can easily extract a sub-subsequence consisting only of terms from A or only of terms from B if we wish.

(2) To show that the union of two sequentially compact sets in a metric space (M,d) is sequentially compact, we need to show that every sequence in the union has a convergent subsequence. Let A and B be two sequentially compact subsets of M, and let X be a sequence in A ∪ B. By (1), X has a subsequence X' that is either in A or in B.

If X' is in A, then it has a convergent subsequence by the sequential compactness of A. This subsequence is also a subsequence of X and therefore converges in A ∪ B. If X' is in B, then it has a convergent subsequence by the sequential compactness of B, and we can again argue that this subsequence converges in A ∪ B.

Therefore, every sequence in A ∪ B has a convergent subsequence, and so A ∪ B is sequentially compact.
(1) Since X = (xk) is a sequence in A ∪ B, each term xk is either in A or in B. Divide the terms of X into two subsequences: X_A consisting of terms in A, and X_B consisting of terms in B. At least one of these subsequences must be infinite (since a finite subsequence cannot exhaust the entire sequence X).

Without loss of generality, assume X_A is infinite. Then X_A is a subsequence of X consisting only of terms in A. Let X' = X_A. Then X' is a subsequence of X such that X' is in A. Similarly, if X_B were infinite, we could construct a subsequence X' in B.

(2) To show that the union of two sequentially compact sets in a metric space (M, d) is sequentially compact, we need to show that any sequence in the union has a convergent subsequence.

Let A and B be two sequentially compact sets in (M, d), and let X = (xk) be a sequence in A ∪ B. By part (1), we know that X has a subsequence X' such that either X' is in A or X' is in B. Without loss of generality, assume X' is in A.

Since A is sequentially compact, X' has a convergent subsequence X'' in A. Thus, X'' is a convergent subsequence of X in A ∪ B. Similarly, if X' were in B, we would have a convergent subsequence in B. Therefore, the union A ∪ B is sequentially compact.

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The series [infinity] ∑k=0 k 10 k converges since the terms of the series approach zero as k increases, and the series satisfies the ratio test.

The ratio of successive terms is 10, which is less than 1, indicating that the series converges.

To show this using the digits of π, we can express the series as [infinity] ∑k=0 c k 10 k , where c k represents the kth digit of π. Since the digits of π are bounded and do not increase indefinitely, the terms of the series also approach zero.

Additionally, the ratio of successive terms can be expressed as c k+1 / c k 10, which is less than 1 for all k, indicating convergence. Therefore, the series [infinity] ∑k=0 k 10 k converges, both in general and when expressed using the digits of π.

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