Show that the series sum=0 to infinity of (2n+1)^3/(n^3+1)^2 converges or diverges
Answers
By the limit comparison test, we can conclude that the original series:
∑\((2n+1)^(3)/(n^(3)+1)^(2)\) also converges.
To determine whether the series ∑(\(2n+1)^(3)/(n^(3)+1)^(2\)) converges or diverges, we can use the limit comparison test. Let's define a new series:
\(b(n) = 1/(n^(3))\)
Now, let's calculate the limit as n approaches infinity of the ratio between the terms of the original series and the terms of the new series:
lim(n→∞) \([(2n+1)^(3)/(n^(3)+1)^(2)] / (1/(n^(3)))\)
To simplify the expression, we'll multiply the numerator and denominator by \((n^(3)+1)^(2):\)
lim(n→∞)\([(2n+1)^(3) * (n^(3)+1)^(2)] / (n^(3))\)
Expanding the numerator:
lim(n→∞) \([8n^(3) + 12n^(2) + 6n + 1] / (n^(3))\)
As n approaches infinity, the terms involving \(n^3\) become dominant, and other terms become negligible in comparison. We can ignore the terms with lower degrees of n:
lim(n→∞) \([8n^(3)] / (n^(3))\)
Simplifying further:
lim(n→∞) 8
Since the limit is a finite nonzero value (8), the original series and the new series b(n) have the same behavior. The series b(n) is a p-series with p = 3, which is known to converge. Therefore, by the limit comparison test, we can conclude that the original series:
∑\((2n+1)^(3)/(n^(3)+1)^(2)\) also converges.
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Related Questions
In a town in Canada, the temperature was 20 °F at sunset. The temperature decreases by 2 every hour. How many hours does it take for the temperature
to fall from 20 °F to -6 °F?
AO 3 hours
B.O 7 hours
C. 13 hours
D. 14 hours
Answers
Answer:
C. 13
Step-by-step explanation:
The drama and math club scenario is represented by the system of equations in which x represents the drama club members and y represents the math club members. x + y = 72 1 4 x + 1 8 y = 14 What number can you multiply the second equation by to create the additive inverse, or the opposite of the y-variable? How many members does the drama club have? How many members does the math club have?
Answers
Answer:
-8, 40, 32.
Step-by-step explanation:
On edge owo <3~
Answer:
-8, 40, 32.
Step-by-step explanation:
The table and graph both represent the same relationship. Which
equation also represents that relationship?
PLEASE someone HELP
Answers
Answer:
y=x^2
Step-by-step explanation:
An algorithm will be used to identify the maximum value in a list of one or more integers. Consider the two versions of the algorithm below. Algorithm I: Set the value of a variable max to - 1. Iterate through the list of integer values. If a data value is greater than the value of the variable max, set max to the data value. Algorithm II : Set the value of a variable max to the first data value. Iterate through the remaining values in the list of integers. If a data value is greater than the value of the variable max, set max to the data value. Which of the following statements best describes the behavior of the two algorithms? A Both algorithms work correctly on all input values. В Algorithm I always works correctly, but Algorithm II only works correctly when the maximum value is not the first value in the list. Algorithm Il always works correctly, but Algorithm I only works correctly when the maximum value is greater than or equal to - 1. D Neither algorithm will correctly identify the maximum value when the input contains both positive and negative input values.
Answers
Algorithm Il always works correctly, but Algorithm I only works correctly when the maximum value is greater than or equal to - 1
=====================================================
Explanation:
Let's say we have the data set {-4,-3,-2}. The value -2 is the largest.
If we follow algorithm 1, then the max will erroneously be -1 after all is said and done. This is because the max is set to -1 at the start even if -1 isn't in the data set. Then we see if each data value is larger than -1.
-4 > -1 is false-3 > -1 is false-2 > -1 is falseEach statement being false means we do not update the max to its proper value -2. It stays at -1.
This is why we shouldn't set the max to some random value at the start.
It's better to use the some value in the data set to initialize the max. Algorithm 2 is the better algorithm. Algorithm 1 only works if the max is -1 or larger.
2. What is the average salary offered to a Stony Brook college graduate? To study this question you and a friend interview N students that graduated last year, and ask them what they earn. Student i's response was recorded as Yi. You are interested in the average, My. You assumed that the sample of Y's is iid. First you calculate the following estimate of uy: W1 N 1 N i=1 ΣΥ. . You and your friend each collected half the data. Thus you collected Y1, ..., YN/2 and your friend collected Yn/2+1, ... , Yn. Unfortunately, it turns out that your friend collected the data at a wild alumnae party, and you suspect that these data may not be as precise as your data. So whereas the variance of your data is, var(Y;) = 0%, i = 1, ...,N/2. then your friends data have the variance, var(Y) = oʻ(1 + 3c), i=N/2+1, ...,N, for some constant c> 0. (d) Your friend is sorry that half the data are not as precise as they could have been, and suggest that you discard the noise data, and simply use hr Na Ex? Y; as your estimator for my. Which estimator is most efficient (has the smallest variance) în or îz? Does your answer depend on c? = N.Σ. (Υ – μ.) - = (e) Suppose now that c = 0 such that var(Y;) = o2 for i = 1, ...,N. You have N = 300 observation and calculate s2 = 20,000,000 and î1 = $48,000. Before collecting the data, your friend argues mean salary, my, is s $50,000, using a 1% significance level. Write down the confidence interval at 1% significance level and decide whether you will accept your friend's
Answers
The more precise estimator ẏ₁ is the most efficient in estimating the average salary. With given values, the confidence interval is calculated to determine whether to accept the claim of a $50,000 mean salary.
In this scenario, we have two estimators for the average salary: ẏ₁, which uses precise data, and ẏ₂, which includes less precise data. The efficiency of the estimators depends on the variance of the data. If we compare the variances, Var(ẏ₁) = 0% and Var(ẏ₂) = o²(1 + 3c). Since Var(ẏ₁) is zero, it implies that ẏ₁ is the most efficient estimator. The answer does not depend on the value of c.
In the second part, with c = 0, we have Var(Y) = o². Given N = 300, s² = 20,000,000, and ẏ₁ = $48,000, we can use these values to construct a confidence interval. Using a 1% significance level, the critical value is 2.57 (from the standard normal distribution). The confidence interval is given by ẏ₁ ± 2.57 * sqrt(s²/N), which results in $48,000 ± 2.57 * sqrt(20,000,000/300). If this interval contains $50,000, we would accept your friend's claim; otherwise, we would reject it.
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Rocky Mountain Tire Center sells 10,000 go-cart tires per year. The ordering cost for each order is $35, and the holding cost is 40% of the purchase price of the tires per year. The purchase price is $25 per tire if fewer than 200 tires are ordered,$17 per tire if 200 or more, but fewer than 8,000, tires are ordered, and $13 per tire if 8,000 or more tires are ordered.
a) How many tires should Rocky Mountain order each time it places an order?
b) What is the total cost of this policy?
Answers
a) Rocky Mountain should order 200 tires each time it places an order.
b) The total cost of this policy is $17,160.
a) To determine how many tires Rocky Mountain should order each time, we need to consider the different price levels and find the point where it is most cost-effective to order. Let's analyze the three price levels:
If fewer than 200 tires are ordered: The purchase price is $25 per tire.
If 200 or more, but fewer than 8,000 tires are ordered: The purchase price is $17 per tire.
If 8,000 or more tires are ordered: The purchase price is $13 per tire.
Since the ordering cost is $35 per order, it is most cost-effective to order the maximum quantity that falls within the second price level, which is 200 tires.
b) To calculate the total cost of this policy, we need to consider the ordering cost and the holding cost. The holding cost is 40% of the purchase price per tire per year. Let's calculate the total cost:
Total holding cost = (Purchase price per tire * Quantity ordered * Holding cost rate) / 2 = (($17 * 10,000 * 0.4) / 2) + (($13 * 2,000 * 0.4) / 2) = $34,000 + $5,200 = $39,200
Total cost = Total ordering cost + Total holding cost = (Ordering cost per order * Number of orders) + Total holding cost = ($35 * (10,000 / 200)) + $39,200 = $1,750 + $39,200 = $40,950
Therefore, the total cost of this policy is $40,950.
Rocky Mountain Tire Center should order 200 tires each time it places an order, resulting in a total cost of $40,950 for this policy. This ordering quantity and cost analysis allows Rocky Mountain to make efficient and cost-effective decisions in managing their inventory.
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(WILL GIVE BRAINLIEST)The coordinate grid shows a polygon. The figure is dilated
according to the rule: (x, y) → ( 3/2 x, 3/2'y). What is the
location of E'?
ANSWER CHOICES:
(4/3,3/4)
(2,8/3)
(9/2,6)
(6,9/2)
Answers
Answer:
i think it’s C
Step-by-step explanation:
i just did 3 times 3/2 then 4 times 3/2 i could be wrong cs i’m in 7th but that’s my best guess
help me please…………………
Answers
Answer:
look at the photo.............
Homework: Section 11.1 Question 7. Complete the square to find the x-intercepts of the function given by the equation listed. f(x)=x² +34x+104 What are the x-intercepts? **** (Simplify your answer. T
Answers
Answer:
x² + 34x + 104 = 0
x² + 34x = -104
x² + 34x + ((1/2)(34))² = -104 + ((1/2)(34))²
x² + 34x + 17² = -104 + 17²
x² + 34x + 289 = 185
(x + 17)² = 185
x + 17 = +√185
x = -17 + √185
What is the big o notation for the following fragment: for (cnt3 = 0, i = 1; i <= n; i *= 2) for (j = 1; j <= n; j++) cnt3++;
Answers
The big O notation for the given code fragment with nested loops is O(n log n), indicating that the runtime grows proportionally to the product of the input size and its logarithm.
The big O notation for the given fragment can be determined by analyzing the nested loops.
The outer loop runs for every power of 2 from 1 to n, which is log₂(n) iterations.
The inner loop runs for n iterations regardless of the value of n.
Therefore, the total number of iterations can be calculated as the product of the iterations of the outer and inner loops, which is O(n log n).
In big O notation, the dominant term is the one that grows fastest as the input size (n) increases. In this case, the log n term dominates over the linear term, so the big O notation for the fragment is O(n log n).
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Simplify.
(8y)2
Write your answer without parentheses.
x
Ś
?
Answers
Answer:
16y
Step-by-step explanation:
(8y)² equals 64y².
Given that an expression, (8y)², we need to simplify it,
Using the concept of Exponents and Powers,
Mathematics uses exponents and powers to indicate repeated multiplication or raising a quantity to a specific power.
They are essential ideas in mathematics and are applied frequently in many disciplines, such as physics, calculus, and algebra.
You must apply the exponent to both the 8 and the y in order to simplify (8y)2 without using parenthesis.
(8y)² = 8² x y²
Now, break each section out separately:
8² = 8 x 8 = 64
Y² stays the same.
As a result, (8y)² becomes 64y².
Hence (8y)² = 64y²
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Which of the following is the solution of the quadratic equation xଶ 3x − 10 0?
Answers
The solutions of the quadratic equation x^2 + 3x - 10 = 0 are x = 4 and x = -1.
The solution of the quadratic equation x^2 + 3x - 10 = 0 can be found by using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Where (a, b, and c) are 1, 3, and -10
So, the solutions of the equation x^2 + 3x - 10 = 0 are:
x = (-3 ± √(3^2 - 4(1)(-10)) ) / 2(1)
x = (-3 ± √(9 + 40)) / 2
x = (-3 ± √49) / 2
x = (-3 ± 7) / 2
x = (4, -1)
Complete question:
Which of the following is the solution of the quadratic equation x^2 - 3x − 10 = 0?
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A regular polygon has exterior angles of 60°
What is the sum of the polygon’s interior angles?
Answers
Answer:
720°
Step-by-step explanation:
The sum of the exterior angles of a polygon = 360°
Divide by 60 to find number of sides (n)
n = 360° ÷ 60° = 6
The sum of the interior angles of a polygon is
sum = 180° (n - 2) ← n is the number of sides
Here n = 6 , then
sum = 180° × 4 = 720°
The sum of the exterior angles of a regular polygon is 360º
Each exterior angle of a regular polygon is 60º/n
360º/n=60º
360/60=n
6=n
A polygon with 6 sides is a hexagon.
Use the formula (n-2)×180
(6-2)*180=4*180=720º
another solution...
you have 6 interior angles (hexagon)
if an exterior angle is 60º, the corresponding interior angle is 180-60=120º
you have 6 of these 120º angles
6*120=720º
evaluate the triple integral. 16y dv, where e is bounded by the planes x = 0, y = 0, z = 0, and 2x 2y z = 4 e
Answers
The value of the triple integral is -16.
Triple integral is a mathematical concept used in calculus to calculate the volume of three-dimensional objects. It extends the concept of a single integral to functions of three variables and integrates over a region in three-dimensional space.
The triple integral of a function f(x, y, z) over a region E in three-dimensional space is denoted by:
∭E f(x, y, z) dV
We can set up the triple integral as follows:
∫∫∫ 16y dV
Where the limits of integration are:
0 ≤ x ≤ 2
0 ≤ y ≤ (2- \(x^2\)z)/(2y)
0 ≤ z ≤ 2/\(x^{2y\)
Note that the upper bound of integration for y is not a constant, but depends on both x and z.
Integrating with respect to y first, we get:
∫∫∫ 16y dV = ∫0^2 ∫\(0^(2-x^2z)/(2x)\)∫\(0^(2/x^2y) 16y dz dy dx\)
= ∫\(0^2\) ∫\(0^(2-x^2z)/(2x) 32/x dx dz\)
= ∫\(0^2\) [16(\(2-x^2z)/x^2\)] dz
= ∫\(0^2 (32/x^2 - 16z)\) dz
= 32∫\(0^2 x^-2 dx - 16\)∫\(0^2\)z dz
= 16 - 16(2)
= -16
Therefore, the value of the triple integral is -16.
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4400 dollars is placed in an account with an annual interest rate of 7%. To the nearest year, how long will it take for the account value to reach 10100 dollars?
Answers
Answer:
19 years approx
Step-by-step explanation:
Given data
Principal P= $4400
Rate r=7%
Final amount A=$10100
Time t= ???
Let us apply the simple interest expression
A=P(1+rt)
substitute
10100=4400(1+0.07*t)
open bracket
10100=4400+308t
10100-4400=308t
5700=308t
t= 5700/308
t= 18.50
Hence the time is 19 years approx
Which of these triangles are translations of Triangle A?
Select all that apply.
А
E
D
B
F
С
please help
Answers
Answer:
B and D
Step-by-step explanation:
A translation of a triangle is moving it up down or side to-side on the axis, not changing the shape flipping it or the orientation,so the translations of a would be B and D!
Hoped this helped, have a good one!
Answer:
B, D
Step-by-step explanation:
a. Previously, we found the series for f(x)= 1+x
1
to be: 1+x
1
=∑ n=0
[infinity]
(−1) n
x n
and its interval of convergence to be (−1,1). Use this series to find a power series for ln(1+x). Explain what process you will use to do this. Write the answer in summation notation. b. Although the radius of convergence of this series is still 1 , you need to test the endpoints (that is, x=1,x=−1 ) to see if they are included in the interval of convergence. c. Use the series in (a) to approximate ln(1.2) with 3 decimal place accuracy.
Answers
To find a power series representation for ln(1+x) using the series for
f(x) = 1+x¹, we will integrate the series term by term.
The resulting series will have the same interval of convergence as the original series. We will then test the endpoints of the interval to determine if they are included in the interval of convergence. Finally, we will use the obtained series to approximate ln(1.2) with 3 decimal place accuracy.
(a) To find the power series representation for ln(1+x), we will integrate the series for f(x) = 1+x term by term.
The series for f(x) is given as:
f(x) = ∑ (-1)ⁿ * xⁿ
Integrating term by term, we get:
∫ f(x) dx = ∫ ∑ (-1)ⁿ * xⁿ dx
= ∑ (-1)ⁿ * ∫ xⁿ dx
= ∑ (-1)ⁿ * (1/(n+1)) * x⁽ⁿ⁺¹⁾ + C
= ∑ (-1)ⁿ * (1/(n+1)) * x⁽ⁿ⁺¹⁾ + C
This series represents ln(1+x), where C is the constant of integration.
(b) The radius of convergence of the obtained series remains the same, which is 1.
To determine if the endpoints x=1 and x=-1 are included in the interval of convergence, we substitute these values into the series. For x=1, the series becomes:
ln(2) = ∑ (-1)ⁿ * (1/(n+1)) * 1⁽ⁿ⁺¹⁾ + C
= ∑ (-1)ⁿ * (1/(n+1))
Similarly, for x=-1, the series becomes:
ln(0) = ∑ (-1)ⁿ * (1/(n+1)) * (-1)⁽ⁿ⁺¹⁾ + C
= ∑ (-1)ⁿ * (1/(n+1)) * (-1)
Since the alternating series (-1)ⁿ * (1/(n+1)) converges, both ln(2) and ln(0) are included in the interval of convergence.
(c) To approximate ln(1.2) using the obtained series, we substitute x=0.2 into the series:
ln(1.2) ≈ ∑ (-1)ⁿ * (1/(n+1)) * 0.2⁽ⁿ⁺¹⁾ + C
By evaluating the series up to a desired number of terms, we can approximate ln(1.2) with the desired accuracy.
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If y − 4 = 2x, which of the following sets represents possible inputs and outputs of the function, represented as ordered pairs? Which order pair is it? {(2, 1), (4, 2), (6, 3)} {(1, 2), (2, 4), (3, 6)} {(4, 0), (6, 1), (8, 2)} {(0, 4), (1, 6), (2, 8)}
Answers
Answer:
the fourth set
Step-by-step explanation:
if you plug this equation into desmos and look at the x-values, you will see that the fourth set is the only one with points that make this equation true.
Answer:
D
Step-by-step explanation:
Darcy keeps $5,000 in a bank account that does
not earn any interest. Every week, she
withdraws 10 percent of the money in the
account for spending. To the nearest cent, how
much has she withdrawn in total after 8 weeks?
A) $2,610.78
B) $2,638.05
C) $2,755.28
D) $2,847.66
Answers
Answer:
Amount withdrawal = $2,847.66 (Approx)
Step-by-step explanation:
Given:
Amount deposit = $5,000
Withdraws percent = 10 % = 0.1
Number of weeks = 8
Find:
Amount withdrawal
Computation:
Amount withdrawal = 5,000[1-(0.1)⁸]
Amount withdrawal = 5,000[0.56953279]
Amount withdrawal = $2,847.66 (Approx)
A new car is valued at $28,000. It increases 9% per year. What is the total value of the car two year after purchase?
Answers
Answer:
$33,040
Step-by-step explanation:
Hope this helps and have a wonderful day!!!
Answer:
$30520
Step-by-step explanation:
in the adjoining fig ABCD is a square and PQRS is a circle if ab=14cm.find the area of shaded portion
Answers
Answer:
Area of shaded proton =42cm²
solve the equation by square roots.
x2\4+7=-9
Answers
The solutions to the equation x²/4 + 7 = -9 are x = 8i and x = -8i, in the complex number system.
To solve the equation x²/4 + 7 = -9 using square roots, first isolate the x² term.
Step 1: Subtract 7 from both sides of the equation:
x²/4 = -16
Step 2: Multiply both sides by 4 to eliminate the fraction:
x² = -64
Since we cannot have a square root of a negative number in real numbers, there is no real number solution for this equation. The equation would have a solution in the complex number system, where the square root of a negative number is expressed using imaginary units (i).
Step 3: Take the square root of both sides:
x = ±√(-64) = ±8i
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Allison's math teacher asked her to write an irrational and rational number. She wrote the following numbers √46 Rational and 5.25 Irrational. Explain if you agree with Allison's identification of rational and irrational numbers.
If you disagree, provide an example of an irrational and rational number.
Answers
An example of a rational number would be √81. This would be rational because this is a perfect square; 81 / 9 = 9, so √81 = 9.
An example of an irrational number would be “7.2534910552…”. This number would be irrational because there’s no repeating pattern within the decimal and it doesn’t terminate.
translations can be moved along an vector at any time and still result in the same image true or false ?
Answers
Answer:
True................
Answer:
True
Step-by-step explanation:
The leg in the right triangle below are each 12 units long
Answers
Answer:
the hypotenuse is 6 and the other leg is 3\(\sqrt{2\)
Step-by-step explanation:
Please help me with my hw 30 points. Correct answer and good explaniation gets brainliest!!
33,34,36,37
Answers
Answer:
35
Step-by-step explanation:
33+34+36+37=140
140/4=35
hope this helps!
In order to determine an interval for the mean of a population with unknown standard deviation a sample of 58 items is selected. The mean of the sample is determined to be 23. The number of degrees of freedom for reading the t value is a. 35.b. 57.c. 36.d. 58.
Answers
If in order to determine an interval for the mean of a population with unknown standard deviation a sample of 58 items is selected. The number of degrees of freedom for reading the t value is .b. 57.
How to find the number of degrees of freedom?Using this formula to find the number of degrees of freedom
Number of degrees of freedom= Number of observations- 1
Where:
Number of observations =58
Let plug in the formula
Number of degrees of freedom= 58 -1
Number of degrees of freedom= 57
Therefore the correct option is b.
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alan can type a research paper in 6 hours. with the help of steve, the paper can be typed in 4 hours. find how long it would take steve to type the paper alone.
Answers
It will take 12 hours to type the research paper by the Steve alone.
What is meant by time and work?Work Done = Time Taken x Rate of Work Work Rate = 1 / Time Taken Time taken = 1 / Rate of work If a piece of work is completed in x days, then the work completed in one day equals 1/x.
The fundamental concept of Time and Work is similar to that of all Arithmetic topics, namely the concept of Proportionality. When the amount of work done is constant, efficiency is inversely proportional to time.
Let x be the amount of hours Steve can spend typing a research paper.
Steve completes 1/x of a research paper per hour.
Alan completed 1/6 of the paper in an hour.
1/4 = amount of paper completed each hour by both of them
1/x + 1/6 = 1/4
Multiply by 12 on the LCD to get 12 + 2x = 3x x = 12 hours.
It will take 12 hours to type the research paper by the Steve alone.
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Some one help pls!!! Solve p = kwm for k
Answers
Can someone please help me ASAP?? It’s due tomorrow!! I will give brainliest If It’s correct.
Answers
Answer: To match the shapes produced by the slice through the triangular prism, we need to consider the orientation of the slice relative to the prism. Here are the matching options:
A. Perpendicular to the base: Rectangle
B. Parallel to the base: Triangle with dimensions equal to the base
C. Diagonal from vertex to vertex: Triangle with unknown dimensions