Determine if the series converges absolutely, converges conditionally, or diverges: a) sigma_n=2^infinity (-1)^n ln (n + 2)/ln (2n) (b) sigma_n=1^infinity (-1)^n 1/Squareroot n (c) sigma_n=1^infinity (-1)^n (1/2)^n Use the Alternating Series Approximation Theorem to find N so that sigma_n=1^N (-1)^n - 1/n^3 gives the sum of the series sigma_n=1^infinity (-1)^n-1/n^3 with less than 000001 error
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a) The series σ(n=2 to ∞) (-1)\(^n\\\) ln((n + 2)/ln(2n)) converges conditionally.
b) The series σ(n=1 to ∞) (-1)\(^n\) 1/√n converges absolutely.
c) The series σ(n=1 to ∞) (-1)\(^n\) (1/2)^n converges.
d) Choose N = 101 to ensure that the partial sum σ(n=1 to N) (-1)^n - 1/n³ has an error less than 0.000001.
a) To determine if the series σ(n=2 to ∞) (-1)^n ln((n + 2)/ln(2n)) converges absolutely, converges conditionally, or diverges, we can use the alternating series test. The alternating series test states that if a series has the form σ(-1)^n b_n where b_n is a positive, non-increasing sequence, then the series converges.
In this case, we have b_n = ln((n + 2)/ln(2n)). To check if b_n is a positive, non-increasing sequence, we can take the derivative:
b_n' = [(1/(n + 2)) * ln(2n) - (ln(n + 2) * 2)] / (ln(2n))².
Since b_n' is negative for n ≥ 2, b_n is a positive, non-increasing sequence. Therefore, the series σ(n=2 to ∞) (-1)^n ln((n + 2)/ln(2n)) converges by the alternating series test.
b) The series σ(n=1 to ∞) (-1)^n 1/√n is an alternating series. To determine if it converges absolutely or conditionally, we can examine the series without the alternating signs, which is σ(n=1 to ∞) 1/√n.
Since the p-series with p = 1/2 (σ(n=1 to ∞) 1/n^(1/2)) converges, the series σ(n=1 to ∞) 1/√n also converges. Therefore, the series σ(n=1 to ∞) (-1)^n 1/√n converges absolutely.
c) The series σ(n=1 to ∞) (-1)^n (1/2)^n is an alternating series. Since the terms of the series approach zero as n goes to infinity, the alternating series test tells us that the series converges.
d) To find N such that the partial sum σ(n=1 to N) (-1)^n - 1/n³ of the series σ(n=1 to ∞) (-1)^n - 1/n³ has an error less than 0.000001, we can use the Alternating Series Approximation Theorem. According to the theorem, the error is bounded by the absolute value of the first omitted term.
In this case, the first omitted term is 1/(N+1)³. Setting this term to be less than 0.000001, we have:
1/(N+1)³ < 0.000001
Solving this inequality, we find N > 100.
Therefore, choosing N = 101 will ensure that the partial sum σ(n=1 to N) (-1)^n - 1/n³ has an error less than 0.000001.
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Related Questions
If there are 12 dogs and 42 cats at a pet daycare, fill out all of the possible ratios of
dogs to cats that could be made.
There are 12 dogs for every 42 cats (12:42 ratio)
There are
dogs for every cats try
Type in an equivalent ratio of dogs and cats.
Answers
There are 2 dogs for every 7 cats. The representative ratio of this would be (2:7 ratio).
Order the rational numbers from least to greatest
Answers
Answer:
its the 3rd one
Step-by-step explanation:
you find which one is smaller and then you go to the next one up
write the equation of the line in fully simplified slope-intercept form.
Answers
Answer:
y=3x-7
Step-by-step explanation:
y=m*x + b : equation of a line.
m= slope = rise over run = 3/1 =3
b = y-axis intercept = -7 (where the line crosses the y-axis)
Answer these please!!!
Answers
The graph of 3x-2y≤6 is the third graph, for 3x-2y<6 is the first graph, for 3x-2y>6 is the fourth graph and for 3x-2y≥6 is the second graph. The solution has been obtained using concept of linear inequality.
What is linear inequality?
A linear inequality is one that would produce a linear equation if the equals relation were used instead of the inequality. When multiplying or dividing both sides by a negative number in order to solve the inequality, the direction of the inequality is reversed. The entire set of solutions to an inequality is known as the solution set.
We are given for graphs, of which two graphs are dotted and two are simple straight line graphs.
The dotted graphs are drawn for the inequalities having < or >
Whereas the simple straight line graphs are drawn for the inequalities having ≤ or ≥.
Now, to notice the shaded pattern, we will see whether the equations are true for (0,0) or not
1. 3x-2y≤6
⇒ 0≤6
So, the equation is true for the point.
Hence, the third graph represents this equation.
2. 3x-2y<6
⇒ 0<6
So, the equation is true for the point.
Hence, the first graph represents this equation.
3. 3x-2y>6
⇒ 0>6
So, the equation is false for the point.
Hence, the fourth graph represents this equation.
4. 3x-2y≥6
⇒ 0≥6
So, the equation is false for the point.
Hence, the second graph represents this equation.
Hence, the graphs are matched with the inequalities.
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Since, there are multiple questions so, the question answered above is attached below.
Twenty-five households are surveyed and the number of children is recorded. The data is summarized in the frequency table below. Use the frequency table to compute the mean of the observations.
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An average mean of 1.72 children can be found on every house
What is frequency table?One method for making data more comprehensible is to put it in a frequency distribution table. Consider having a list of IQ results for a talented class at a certain elementary school. The frequency of a value in a set of data is how frequently it appears.
Victor, for instance, made nine attempts to obtain a red gumball. The quantity of gumballs in each hue that were produced would be the frequency in this situation. The distribution of observations based on a variable's possible values is displayed in a frequency table.
frequency = 10 + 4 + 2 + 3 + 4 + 2
= 25
Sum of children
= (10 × 0)+(4 ×1)+(2 × 2)+(3 × 3)+(4 × 4)+(2 × 5)
= 43
Mean = 43/25
= 1.72
Thus, An average mean of 1.72 children can be found on every house.
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Which Triangle Below has the greatest area?
Answers
Answer:
number 1
Step-by-step explanation:
Given the regression equation y-hat = 15.6 - 3.8x, the predicted y for x = 3 is ___________.
Answers
Work Shown:
y = 15.6 - 3.8x
y = 15.6 - 3.8*3
y = 4.2
According to recent data, women make up what percentage of workers in science and technology (STEM) fields in Canada and the United States, respectively?
A. 34% and 40%
B. 23% and 26%
C. 17% and 26%
D. 25% and 27%
E. 34% and 26%
Answers
According to recent data, women make up 34% and 26% of workers in science and technology (STEM) fields in Canada and the United States, respectively. The correct option is A. 34% and 26%.
According to recent data, women make up 34% and 26% of workers in science and technology (STEM) fields in Canada and the United States, respectively. This indicates that women are still underrepresented in STEM fields, despite the fact that there has been an effort to attract more women to STEM fields.
In both Canada and the United States, women have made significant progress in breaking down gender barriers in STEM fields. However, there is still work to be done to close the gender gap and increase representation of women in STEM fields.
Women's representation in STEM fields has increased in both Canada and the United States in recent years, but the percentage of women in STEM fields is still significantly lower than the percentage of men. More efforts are needed to close the gender gap in STEM fields and encourage more women to pursue STEM careers.
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Please help with the 2nd one
Answers
Answer:
Step-by-step explanation:
1807
a tile store charges $747.50 to install 115 square feet of tile. assuming they charge the same rate per square foot regardless of the amount of tile installed, how much would they charge to install 970 square feet of tile?
Answers
With the help of simple mathematical operations, we know that the cost of installing 970 ft² will be $6,305.
What are mathematical operations?A method of calculation: The most popular ones are (+, −, ×, ÷) for add, subtract, multiply, and divide.
However, there are many more, like square roots, logarithms, and squares.
It is most likely an operation if it isn't a number.
Example: The operation in 25 + 6 = 31 is added.
So, we know that:
The store charges $747.50 to install 115 ft² of tiles.
Then, the amount that will come to install 970 ft² tiles:?
Cost of installing 1 ft² of tiles:
747.50/115
$6.5
So, the cost of installing 1 ft² of tiles: is $6.5.
Now, the cost of installing 970 ft² tiles will be:
6.5 × 970
$6,305
Therefore, with the help of simple mathematical operations, we know that the cost of installing 970 ft² will be $6,305.
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Use the graph to find the average rate of change from x=-1 to x=1.
Answers
The average rate change of over the interval [-1,1] is 2.
What is an interval?
In mathematics, an interval is expressed in numerical terms. All the numbers between two specific numbers are referred to as an interval. All actual numbers between those two are included in this range.
Given points on the graph is
x -2 -1 0 1
y -4 -3 -1 3
The average rate of change of a function is defined by the expression over the interval a ≤ x ≤ b,
Average rate of change = [f(b) - f(a)]/(b - a)
Here a = -1, b = 1, f(a) = -3, and f(b) = 3
The average rate is (3 - (-3))/ (1 - (-1)) = 2
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What type of association does the graph show between x and y?
Linear positive association
Nonlinear positive association
Linear negative association
Nonlinear negative association
Answers
Answer:
Linear negative association
Step-by-step explanation:
All points are in a line, so it is linear.
As x increases, y decreases. That is a negative association.
Answer: Linear negative association
find all values of x such that (6, x, −11) and (5, x, x) are orthogonalfind all values of x such that (6, x, −11) and (5, x, x) are orthogonal
Answers
The values of x that make the vectors (6, x, -11) and (5, x, x) orthogonal are x = 5 and x = 6.
To find all values of x such that (6, x, -11) and (5, x, x) are orthogonal, we will use the dot product of the two vectors. Two vectors are orthogonal if their dot product is zero.
2 vectors are called orthogonal if they are perpendicular to each other, and after performing the dot product analysis, the product they yield is zero.
In mathematical terms, the word orthogonal means directed at an angle of 90°. Two vectors u,v are orthogonal if they are perpendicular, i.e., they form a right angle, or if the dot product they yield is zero.
1: Write down the two vectors.
Vector A = (6, x, -11)
Vector B = (5, x, x)
2: Calculate the dot product of the two vectors.
A · B = (6 * 5) + (x * x) + (-11 * x)
3: Set the dot product equal to zero, as the vectors are orthogonal.
30 + x^2 - 11x = 0
4: Rearrange the equation to solve for x.
x^2 - 11x + 30 = 0
5: Factor the quadratic equation.
(x - 5)(x - 6) = 0
6: Solve for x.
x = 5 or x = 6
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What number is a square root of 1 ?
Answers
Answer:
1 is the square root of 1.
Hope that helps! :)
Step-by-step explanation:
Two cards are randomly chosen from a standard deck of cards with replacement. What is the probability of successfully drawing, in order, a three and then a queen?
Answers
The probability of successfully drawing a three and then a queen in order, with replacement, is 1/169.
To calculate the probability of successfully drawing a three and then a queen, we need to consider the number of favorable outcomes divided by the total number of possible outcomes.
In a standard deck of cards, there are 4 threes and 4 queens. Since replacement is allowed, the probability of drawing a three on the first draw is 4/52 (4 favorable outcomes out of 52 total cards).
After successfully drawing a three, there are still 52 cards left in the deck, including 4 queens.
Therefore, the probability of drawing a queen on the second draw is 4/52.
To find the probability of both events occurring in order, we multiply the probabilities together:
P(3 and then queen) = P(3) * P(queen) = (4/52) * (4/52) = 16/2704 = 1/169
So, the probability of successfully drawing a three and then a queen in order, with replacement, is 1/169.
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Line I1 and I2 are parallel lines cut by a transversal t. Write the angle relationship for each pair what is
Answers
Answer:
Step-by-step explanation:
∠b & ∠g are consecutive exterior angles
∠e & ∠c are alternate interior angles.
∠b and ∠f are corresponding angles.
∠h and ∠c are supplementary angles.
5 gallons to 6 quarts in simplest ratio
Answers
The simplest ratio between 5 gallons and 6 quarts is 10:3.
To convert 5 gallons to quarts, we can use the conversion factor that 1 gallon is equal to 4 quarts.
Therefore, 5 gallons is equal to 5 x 4 = 20 quarts.
Now we need to express this ratio in its simplest form.
To simplify the ratio, we can divide both the number of quarts and the number of gallons by their greatest common divisor (GCD).
The GCD of 20 and 6 is 2. So, dividing both numbers by 2, we get:
20 ÷ 2 = 10
6 ÷ 2 = 3
Therefore, the simplest ratio between 5 gallons and 6 quarts is 10:3.
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Work out(−3)2−3×(2×54)÷112
Answers
Answer:
-249/28
Step-by-step explanation:
use the probability rules in this chapter to solve each of the following. (a) suppose in 2012, the probability that a randomly selected child in a country was living with his or her mother as the sole parent was 0.242 and with his or her father as the sole parent was 0.080. what was the probability that a child was living with just one parent?
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The probability that a child was living with just one parent is 0.322
The probability that a randomly selected child in a country was living with his or her mother as the sole parent was 0.242
P(mother as sole parent) = 0.242
The probability that a randomly selected child in a country was living with his or her father as the sole parent was 0.080
P(father as sole parent) = 0.080
P(A U B) = P(A) + P(B) - P(A∩ B)
P(living with just one parent) = P(mother as sole parent) + P(father as sole parent)
P(living with just one parent) = 0.242 + 0.080 = 0.322
Therefore, the probability that a child was living with just one parent is 0.322
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PLEASE HELP. Three tennis balls are stored in a cylindrical container with a height of 8.8 inches and a radius of 1.42 inches. The circumference of a tennis ball is 8 inches. Find the amount of space within the cylinder not taken up by the tennis balls. Round your answer to the nearest hundredth.
Amount of space: about ___ cubic inches
Answers
The amount of space within the Cylindrical container not taken up by the tennis balls is approximately 27.86 cubic inches, rounded to the nearest hundredth.
The amount of space within the cylindrical container not taken up by the tennis balls, we need to calculate the volume of the container and subtract the total volume of the three tennis balls.
The volume of the cylindrical container can be calculated using the formula for the volume of a cylinder:
Volume = π * r^2 * h
where π is a mathematical constant approximately equal to 3.14159, r is the radius of the cylinder, and h is the height of the cylinder.
Given that the radius of the cylindrical container is 1.42 inches and the height is 8.8 inches, we can substitute these values into the formula:
Volume of container = 3.14159 * (1.42 inches)^2 * 8.8 inches
Calculating this expression:
Volume of container ≈ 53.572 cubic inches
The volume of each tennis ball can be calculated using the formula for the volume of a sphere:
Volume of a sphere = (4/3) * π * r^3
Given that the circumference of the tennis ball is 8 inches, we can calculate the radius using the formula:
Circumference = 2 * π * r
Solving for r:
8 inches = 2 * 3.14159 * r
r ≈ 1.2732 inches
Substituting this value into the volume formula:
Volume of a tennis ball = (4/3) * 3.14159 * (1.2732 inches)^3
Calculating this expression:
Volume of a tennis ball ≈ 8.570 cubic inches
Since there are three tennis balls, the total volume of the tennis balls is:
Total volume of tennis balls = 3 * 8.570 cubic inches
Total volume of tennis balls ≈ 25.71 cubic inches
Finally, to find the amount of space within the cylinder not taken up by the tennis balls, we subtract the total volume of the tennis balls from the volume of the container:
Amount of space = Volume of container - Total volume of tennis balls
Amount of space ≈ 53.572 cubic inches - 25.71 cubic inches
Amount of space ≈ 27.86 cubic inches
Therefore, the amount of space within the cylindrical container not taken up by the tennis balls is approximately 27.86 cubic inches, rounded to the nearest hundredth.
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A random sample of 121 bottles of cologne showed an average content of 4 ounces. It is known that the standard deviation of the contents (i.e., of the population) is 0.22 ounces. Refer to Exhibit 7-4. The point estimate of the mean content of all bottles is? a) 0.22 b) 4 c) 121 d) 0.02
Answers
The point estimate of the mean content of 121 bottles with the average content of 4 ounces and the standard deviation 0.22 ounces is 4. Option B is the correct answer.
A point estimate is a statistic used to predict the corresponding population parameter. In a sense, a point estimate is a guess about a parameter that uses data from a sample as the basis of the guess. Point estimates can be used to estimate population parameters such as the population mean or population standard deviation.
The point estimate is a single numerical value that represents a guess for a parameter. The sample mean and sample standard deviation are examples of point estimates for the population mean and population standard deviation. If a point estimate is a good guess, it should be fairly close to the true value of the parameter.
Thus, option B is the correct answer.
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The radius of a circle is 12 kilometers. What is the circle's area?
Use 3.14 for π
Please Help
Answers
Answer:
I believe your answer is 452.39
Step-by-step explanation:
Suppose someone claims the average delivery time for u.s. packages to be delivered during december is 2 days. you believe it takes longer than that. you conduct a hypothesis test; what are your hypotheses?
Answers
The null hypothesis (H0) is that the average delivery time for U.S. packages during December is 2 days, while the alternative hypothesis (Ha) is that the average delivery time is longer than 2 days.
Based on your question, you want to test if the average delivery time for U.S. packages in December is longer than 2 days. In this hypothesis test, you would have the following hypotheses:
Null Hypothesis (H0): The average delivery time for U.S. packages in December is 2 days.
Alternative Hypothesis (H1): The average delivery time for U.S. packages in December is greater than 2 days.
Thus, H0: μ ≤ 2 (where μ is the population mean delivery time)
Ha: μ > 2
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Suppose y varies inversely with x, and y = 49 when x = 17
. What is the value of x when y = 7 ?
Answers
Answer:
119 is the value of x when y = 7
Step-by-step explanation:
Since y varies inversely with x, we can use the following equation to model this:
y = k/x, where
k is the constant of proportionality.Step 1: Find k by plugging in values:
Before we can find the value of x when y = k, we'll first need to find k, the constant of proportionality. We can find k by plugging in 49 for y and 17 for x:
Plugging in the values in the inverse variation equation gives us:
49 = k/17
Solve for k by multiplying both sides by 17:
(49 = k / 17) * 17
833 = k
Thus, the constant of proportionality (k) is 833.
Step 2: Find x when y = k by plugging in 7 for y and 833 for k in the inverse variation equation:
Plugging in the values in the inverse variation gives us:
7 = 833/x
Multiplying both sides by x gives us:
(7 = 833/x) * x
7x = 833
Dividing both sides by 7 gives us:
(7x = 833) / 7
x = 119
Thus, 119 is the value of x when y = 7.
Can you please help on 9,10,11
Answers
Answer:
For 8 - 11
8. WX = 16
9. RW = 32
10 WZ = 15
11. QZ = 45
For 12 - 15
12. BD = 7
13. HB = 14
14. CB = 9
15. CK = 27
Step-by-step explanation:
look at the pictures
(BRAINLIEST) Please answer quickly i will mark brainliest if right
Answers
Answer:
D
Step-by-step explanation:
yo u dont even have to work just look that its obtuse and 126 is obtuse so there u go!
Answer:
A seems right to me
Step-by-step explanation:
Tina needs 36 flowers for her next project. The flowers are sold in bunches
of 5.How many bunches will she need?
Answers
Name the ordered pair for the point
Answers
Answer:
the answer is 1,1 1/2 os c
Step-by-step explanation:
Working out is required, please. 30 points!
Answers
Answer:
Volume = 11 cm x 9 cm x 7 cm
Volume = 693 cubic cm I think I'm not sure
Step-by-step explanation:
during a practice sash drank 2 1/2 pints of water and on the way home she drank 1/2 cup of water how much water did she drank in total
Answers
Answer:
5 1/2 cups
Step-by-step explanation:
1 pint = 2 cups
(2 1/2)(2) = 5
5 + 1/2 = 5 1/2