Answers
This is a telescoping sum. The K-th partial sum is
\(S_K = \displaystyle \sum_{k=1}^K \left(\frac1{\sqrt{k+1}} - \frac1{\sqrt{k+3}}\right) \\\\ ~~~= \left(\frac1{\sqrt2} - \frac1{\sqrt4}\right) + \left(\frac1{\sqrt3} - \frac1{\sqrt5}\right) + \left(\frac1{\sqrt4} - \frac1{\sqrt6}\right) + \left(\frac1{\sqrt5} - \frac1{\sqrt7}\right) + \cdots \\\\ ~~~~~~~~+ \left(\frac1{\sqrt{K-1}} - \frac1{\sqrt{K+1}}\right) \\\\ ~~~~~~~~+ \left(\frac1{\sqrt K} - \frac1{\sqrt{K+2}}\right) + \left(\frac1{\sqrt{K+1}} - \frac1{\sqrt{K+3}}\right)\)
\(\displaystyle = \frac1{\sqrt2} + \frac1{\sqrt3} - \frac1{\sqrt{K+2}} - \frac1{\sqrt{K+3}}\)
As \(K\to\infty\), the two trailing terms will converge to 0, and the overall infinite sum will converge to
\(\displaystyle \sum_{k=1}^\infty \left(\frac1{\sqrt{k+1}} - \frac1{\sqrt{k+3}}\right) = \lim_{k\to\infty} S_k = \boxed{\frac1{\sqrt2} + \frac1{\sqrt3}}\)
By the limit comparison test, the expression √[1 / (1 + 1 / k)] - √[1 / (1 + 3 / k)] has a limit, then the expression [1 / √(k + 1)] / [1 /√k] - [1 / √(k + 3)] / [1 /√k] has a limit and the series ∑ [1 / √(k + 1)] - ∑ [1 / √(k + 3)] is convergent.
Is the series convergent?
Herein we have a series that involves radical components. First, we simplify the expression given:
∑ [1 / √(k + 1) - 1 / √(k + 3)] = ∑ [1 / √(k + 1)] - ∑ [1 / √(k + 3)]
The convergence of the series can be proved by the limit comparison test, where each component of the subtraction of the series is compared with a series that is convergent. We notice that both 1 / √(k + 1) and 1 / √(k + 3) resembles the expresion 1 /√k. Then, we have the following subtraction of ratios:
[1 / √(k + 1)] / [1 /√k] - [1 / √(k + 3)] / [1 /√k]
√k / √(k + 1) - √k / √(k + 3)
√[k / (k + 1)] - √[k / (k + 3)]
Then, by using the limit property for rational functions we find the following result for n → + ∞:
√[1 / (1 + 0)] - √[1 / (1 + 0)]
√1 - √1
1 - 1
0
By the limit comparison test, the expression √[1 / (1 + 1 / k)] - √[1 / (1 + 3 / k)] has a limit, then the expression [1 / √(k + 1)] / [1 /√k] - [1 / √(k + 3)] / [1 /√k] has a limit and the series ∑ [1 / √(k + 1)] - ∑ [1 / √(k + 3)] is convergent.
Remark
The statement is incomplete and complete form cannot be found, therefore, we decided to determine if the series is convergent or not.
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Related Questions
I WILL GIVE 30 POINTS AND WILL MARK YOU THE BRAINIEST IF YOU ANSWER THESE 2 QUESTIONS RIGHT!!!!!
1. Find the coordinates of the orthocenter of a triangle with the vertices (5,3), (8,6), (0,14) at each set of points on a coordinate plane?
2. Find the coordinates of the orthocenter of a triangle with the vertices (0,0), (8,2), (2,8) at each set of points on a coordinate plane?
Answers
Answer:
1. (8, 6) 2. (16/5, 16/5) = (3.2, 3.2)
Step-by-step explanation:
I have attached e graphs of the 2 triangles below respectively.
What is the НСF of
4725
5850
Answers
Answer:
433345344 Answer
Step-by-step explanation:
evaluate 2(cos 45°sin 45° + tan²30
Answers
The value of the expression 2(cos 45°sin 45° + tan²30°) is 5/3.
Let's evaluate the given expression :
cos 45° = √2/2 (This is a standard value for cosine of 45 degrees.)
sin 45° = √2/2 (This is a standard value for sine of 45 degrees.)
tan 30° = sin 30° / cos 30° = (1/2) / (√3/2) = √3/3 (This is a standard value for tangent of 30 degrees.)
Now, let's substitute these values back into the original expression:
2(cos 45°sin 45° + tan²30°)
= 2(√2/2 * √2/2 + (√3/3)²)
= 2(1/2 + 3/9)
= 2(1/2 + 1/3)
= 2(3/6 + 2/6)
= 2(5/6)
= 10/6
= 5/3
Therefore, the value of the expression 2(cos 45°sin 45° + tan²30°) is 5/3.
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Select the correct answer.
Mateo teaches a continuing education class at the library on Tuesday nights. He estimates that 75% of his students are satisfied or very satisfied with the class.
Last week, he asked a random sample of students to take a survey on their experience in the class. However, the results showed that only 70% indicated that they are satisfied or very satisfied with the class. He decides to randomly survey more of his students. How will Mateo know whether his model is valid or not?
A.
It is valid if the result from all the surveys and the model get closer together.
B.
It is impossible to determine whether the model is valid.
C.
It is valid if the result from all the surveys and the model stay the same.
D.
It is valid if the result from all the surveys and the model get farther apart.
Answers
Answer:
A
Step-by-step explanation:
Getting closer together means that both are becoming more accurate.
A valid model is such model whose calculated value approximates to the actual value. If it doesn't approximate to the actual value, it must be close to the actual value.
The model is valid if the results from the survey are close enough, together.
Given that:
\(p_1 = 75\%\) -- the actual estimation of satisfied students
\(p_2 = 70\%\) -- the second estimation of satisfied students
For a model to be valid, the calculated value must be close to the actual value.
If after Mateo's survey of more students, the result of satisfied or very satisfied students is still close to 75%, then Mateo's model is valid.
If otherwise, then the model is invalid.
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which part in the steering column allows for changes in the angle between the upper and lower shafts?
Answers
The part in the steering column that allows for changes in the angle between the upper and lower shafts is the universal joint.
The universal joint, also known as a U-joint or Cardan joint, is a mechanical device that enables changes in the angle between two shafts that are not in a straight line. In the context of a steering column, the upper and lower shafts are connected by a universal joint.
As the steering wheel is turned, it exerts rotational force on the upper shaft of the steering column. The universal joint allows for the transmission of this rotational motion while accommodating changes in the angle between the upper and lower shafts. It provides flexibility and compensates for any misalignment between the two shafts, ensuring smooth and continuous rotation of the steering column.
The universal joint typically consists of two yokes connected by a cross-shaped component with needle bearings. These bearings allow for rotation in multiple axes, allowing the steering column to handle variations in the angle between the upper and lower shafts during steering movements.
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Determine whether each of the following functions is a solution of laplace's equation uxx + uyy = 0. (select all that apply. ).
Answers
Solution of laplace´s equation uxx + uyy = 0 including these function :
u(\(x^{2} + y^{2})\) = 0uxx = uyyu(xx) = u(yy)ux(x) = uy(y)u \((x+y)^{2}\) = 0Hence, the solutions of laplace´s equation is including the following functions above.
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represent 3/4 and 8/9 on a number line
Answers
Answer:
Step-by-step explanation:
3/4 and 8/9 both lie between 0 and 1. Their LCD is 36.
Thus, 3/4 = 27/36 and 8/9 = 32/36.
Dividing the number line between 0 and 1 into 36 equal lengths, plot a dot at the 27th such mark and then another dot at the 32nd mark.
This clearly shows that 32/36 is larger than 27/36.
how do i find the value of x?
Answers
Answer:
x = 18 (verified by algebra) ✅Step-by-step explanation:
When we have two similar triangles, we can use a proportion to determine an unknown value.
\(\frac{2x + 80}{87} = \frac{80}{60}\)
We will cross multiply to get two equivalent equations that can be solved.
(2x + 80) x 60 = 120x + 4800
We can simplify this by dividing both sides by 24 to get a much simpler equation to work with: 5x + 200.
80 x 87 = 6960
We will also divide this by 24. The answer is 290
5x + 200 = 290
Subtract 200
5x = 90
Divide by 5
x = 18
Now, we can also check to make sure we did everything correctly.
(2(18) + 80) x 60 = 120x + 4800
(36 + 80) x 60 = 120(18) + 4800
116 x 60 = 2160 + 4800
6960 = 6960 ✅
Peiling and John saved the same amount of money. After Peiling lent $28 to John, John had 5 times as much money as Peiling. How much did John have in the end?
Answers
Answer:
140$
Step-by-step explanation:
so if peilling lent 28$ and then john has 5x as much that means john had 4x as much before getting the 28$ which would mean that john had 112$ dollars so now with that knowledge just add the 28$ to get 140$ or just do 5 x 28 to get 140
\( \binom{4a + 3}{5} - \binom{2a + 3}{3} \)
simplify the fraction
Answers
I assume your question was to simplify the following fraction ;
\({:\implies \quad \sf \dfrac{4a+3}{5}-\dfrac{2a+3}{3}}\)
Taking LCM will yield ;
\({:\implies \quad \sf \dfrac{3(4a+3)-5(2a+3)}{15}}\)
\({:\implies \quad \sf \dfrac{12a+9-10a-15}{15}}\)
\({:\implies \quad \boxed{\bf \dfrac{2a-6}{15}}}\)
This is the required answer
the area of a rectangular room is 45 CM square if the length had been 3 M less and breadth 1 m more it would have been a square find the length and breadth of the room
Answers
The length of this rectangular room is equal to 9 meters.
The breadth of this rectangular room is equal to 5 meters.
How to calculate the area of a rectangle?In Mathematics and Geometry, the area of a rectangle can be calculated by using the following mathematical equation:
A = xy
Where:
A represent the area of a rectangle.y represent the breadth of a rectangle.x represent the length or height of a rectangle.If length would become 3 meters less and breadth would become 1 meter more, we have the following expressions;
y = x - 3
x = y + 1
45 = xy
When the rectangle turns to a square, all of the side lengths become equal;
y + 1 = x - 3
x = y + 4
From the area of a rectangle formula, we have:
45 = (y + 4)y
45 = y² + 4y
y² + 4y - 45 = 0
y² + 9y - 5y - 45 = 0
y(y + 9) - 5(y + 9) = 0
y = 5 or -9 meters.
For the length, we have:
x = y + 4
x = 5 + 4
x = 9 meters.
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What is the GCF of 48m5n and 81m2n2? 3mn 3m2n 48m2n 129m5n2.
Answers
The correct option is B which is the GFC is \(\rm 3m^2n\).
What is GCF?The GCF (Greatest Common Factor) of two or more numbers is the greatest number among all the common factors of the given numbers.
To find the GCF of the two terms, the continuous division must be done.
Therefore,
The GFC is;
\(\rm= \dfrac{81m^5 n }{48 m^2} \times n^2 \\\\= \dfrac{27}{16 }\dfrac{ m^3}{n}\)
Here 3 is the common factor for the coefficient.
So, the common factor is;
\(\rm \dfrac{81}{27}=3\\\\\dfrac{m^5}{m^3}=m^2\\\\\dfrac{n^2}{n}=n\)
Hence, The correct option is B which is the GFC is \(\rm 3m^2n\).
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Answer:
b
Step-by-step explanation:
The coordinates of three vertices of a square A (-2 1/2 , 1 1/2), B (-2 1/2 -3), and C (2,1 1/2 when point D is placed on this square what will the perimeter of the square be?
Answers
Answer:
The perimeter of square will be 18 units
Step-by-step explanation:
We are given that the coordinates of three vertices of a square A (-2 1/2 , 1 1/2), B (-2 1/2 -3), and C (2,1 1/2 )
When point D is placed on this perimeter.
We have to find the perimeter of the square.
First we have to find the side of square by using the distance formula
Distance formula
\(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
Coordinates of A=(-5/2,3/2)
Coordinates of B=(-5/2,-3)
Length of side AB=\(\sqrt{(-5/2+5/2)^2+(-3-3/2)^2}\)
Length of side AB=\(\sqrt{0+(-9/2)^2}\)
Length of side AB=9/2 units
Now, the perimeter of square=4 (side)
Using the formula
Perimeter of square=4(9/2)=18 units
What kind of function is being illustrated by f(x)=|2x³-3x|+5?
Answers
Answer:
Option D: Absolute value function
Step-by-step explanation:
The absolute value of a number is the distance of the number from 0 to the left or right on the number line.
We are given the function;
f(x) = |2x³ - 3x| + 5
This function contains an absolute value symbol which is |2x³ - 3x|.
This function is thus illustrated by an an absolute value function because an absolute value function will be one that contains algebraic expressions within absolute value symbols.
We want to see which type of function
From the comments, we can see that the options are:
a. Rational Function b. Constant Function c. Greatest Integer Function d. Absolute Value FunctionThe correct option is d: Absolute Value Function
To analyze the type of function that we have, we need to see the main operation that acts on the function.
Here we have:
f(x)=|2x³-3x|+5
Note that all our variables are inside the absolute value operation.
Then we can conclude that this an absolute value function.
Then the correct option is d: Absolute Value Function
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how many 1/2 cup serving would 3 gallons of punch provide?
Answers
Answer: 96 servings.
Step-by-step explanation:
There are 16 cups in 1 gallon, so 3 gallons of punch would be equal to:
3 gallons x 16 cups/gallon = 48 cups
If each serving size is 1/2 cup, then the number of servings in 3 gallons of punch would be:
48 cups / (1/2 cup/serving) = 96 servings
Therefore, 3 gallons of punch would provide 96 servings, assuming each serving size is 1/2 cup.
Which of the following is a statistical question with categorical data?
Question 1 options:
A. Do you have five siblings?
B. What color eyes do you have?
C. What is the favorite subject of sixth graders at my school?
D. What is Mrs. Christmas's favorite subject?
Answers
Answer:
A
Step-by-step explanation:
because when we say statical data we mean numerical data (asking u about number) so the answer yes i have 5 or no they r more or less.
15 The table shows values of s and t.
S
t
0.2
7.5
0.5
1.4
0.9
Is s inversely proportional to f? Explain why.
(2 marks)
Answers
Answer:
s is not inversely proportional to t
Step-by-step explanation:
This is an edited response. My first answer was incorrect.s is not inversely proportional to t. I had responded that they were, based on the fact that as s went up, t went down. But the question was not simply is there an inverse relationship, but are they inversely proportional.
The term proportional means that the relationship between s and t is a constant. That is:
t = s*(1/x)
Let's rewrite that to y*x = k and then check the numbers. See the attached spreadsheet. If the relationship were inversel proportioanl, thaen the product of t*s would be a contant for the series. The third set is different from the first two. The data has an is inverse relationship, but it is NOT proportional.
An astronomical unit (AU) is the average distance between Earth and the sun. One AU is equal to approximately 150,000,000 km. Astronomers also use light years to measure distance. A light year is approximately 9.5×10^12
km.
Use scientific notation to rewrite the measure of one astronomical unit (1 AU) in kilometers
Answers
Using scientific notation, it is found that the measure of one astronomical unit in kilometers is \(1.5 \times 10^8\)
A number in scientific notation is given by:
\(a \times 10^b\)
In which \(a \in [1, 10)\)To realize the conversion from decimal to scientific notation, b is the number of decimal places the comma moves, being positive if to the left, negative if to the right.In this problem, 1 AU = 150,000,000 km.
The comma has to move 8 places to the right, hence \(b = 8\)Then, the measure is: \(1.5 \times 10^8\)
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Which expression is equivalent to 2 (a + 2b)- a - 2b?
Answers
Answer:
it's linear expression cuz the highest power of the unknown is 1
Answer:
\(\bf a+2b\)Step-by-step explanation:
\(\tt 2 (a + 2b)- a - 2b\)
Expand:-
\(\tt 2a+4b-a-2b\)
Combine like terms:-
\(\tt 2a-a=\bf a\)
\(\tt a+4b-2b\)
\(\tt 4b-2b=\bf 2b\)
\(\tt a+2b\)
______________________
Hope this helps!
b) The average age of Kishan and Dolma is 15 years and that of Dolma and Shashwat is 14 years. If Kishan is 17 years old, find the age of Dolma and Shashwat.
Answers
Step-by-step explanation:
average means the sum of all values divided by a number of values
The 2kg mass reverses direction after the collision and has a velocity of 3 m/sec. What is the new velocity of the 4kg mass?
Answers
The new velocity of the 4kg mass can be determined using the conservation of momentum equation, which states that momentum (m * v) is conserved before and after a collision.
The momentum of the 2 kg mass prior to the collision is (2 kg) * (5 m/s) = 10 kg m/s. Because the 2 kg mass reverses direction after the collision, its momentum is now -10 kg m/s. Thus, the sum of the two masses and velocities before and after the collision must be equal.
The total momentum before the collision is: 10 kg m/s + 0 kg m/s = 10 kg m/s. And the total momentum after the collision is: -10 kg m/s + (4 kg)Vf = (4 kg)Vf - 10 kg m/s where Vf is the velocity of the 4 kg mass after the collision
Therefore, Vf = (10 kg m/s + 10 kg m/s) / 4 kg = 5 m/s, which means that v = 5 m/sec. Thus, the new velocity of the 4kg mass is 5 m/sec.
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in △ABC, B=51°, b=35, and a=36. what are the two possible values for angle A to the nearest tenth of a degree?
Select all that apply:
a. A = 129.9°
b. A = 53.1°
Answers
Both options a. A = 129.9° and b. A = 53.1° are correct.
To find the possible values for angle A in triangle ABC, we can use the Law of Sines, which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.
Using the Law of Sines, we have sin(A)/a = sin(B)/b. Plugging in the given values, we get sin(A)/36 = sin(51°)/35.
To find the two possible values for angle A, we can solve the equation sin(A)/36 = sin(51°)/35. Taking the arcsine of both sides, we have A = arcsin((sin(51°)/35)*36).
Calculating this expression, we find two possible values for angle A:
A ≈ 53.1° (rounded to the nearest tenth)
A ≈ 129.9° (rounded to the nearest tenth)
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how many solutions is in x + 5 = 0
Answers
Answer:
1 answer which would be -5+5 which =0
Step-by-step explanation:
Find the area of each.
Answers
Answer:
Hope the pictures will help you..
How do I find the area of a trapezoid 6ft by 9ft
Answers
Given the following question:
Where the trapezoid is...
6ft by 9ft
Find the area:
\(A=(a+b)\times h\div2\)What is the range of this relation?
Answers
range is the y-values
Solve the quadratic equation. Show all of your steps.
x² + 3x – 5= 0
Answers
Answer:
\(x = \dfrac{-3 \pm \sqrt{29}}{2}\)
Step-by-step explanation:
x² + 3x – 5= 0
-5 = -5 * 1 = -1 * 5
-5 + 1 = -4
-1 + 5 = 4
To factor, we need two integers that multiply to -5 and add to 3. There are no such integers, so the trinomial is no factorable. We use the quadratic formula.
\(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
\(x = \dfrac{-3 \pm \sqrt{3^2 - 4(1)(-5)}}{2(1)}\)
\(x = \dfrac{-3 \pm \sqrt{9 + 20}}{2}\)
\(x = \dfrac{-3 \pm \sqrt{29}}{2}\)
the random variable x has a gamma distribution with mean 8 and variance 16. what are the values for the shape and scale parameters? use the mean and variance formulas given in the notes. (a) shape
Answers
The shape and scale parameters have values of 4 and 2, respectively.
For Gamma distribution denoted by Gamma(k,θ) here k is the shape parameter and θ is the scale parameter.
What is shape parameter ?
A shape parameter affects the general shape of a distribution, as the name implies; they are a family of distributions with various shapes. The parameters are frequently calculated from recent data or occasionally extrapolated from historical statistical data.
What is scale parameter ?
Graphs have meaning thanks to scale settings. The scale in a typical normal model is equal to the standard deviation,. Even though the area under the graph is 1, you can't take any information from it without a scale. A standard normal distribution without any scaling parameters is shown in the top graph.
The mean is given by the formula
Mean = kθ
And the variance is,
Variance = kθ^2
So now according to the given information we have,
θ=8
kθ^2=16
So dividing the two equations,
θ = 16/8=2
Using this in first equation,
kθ = 8
2k=8
k=4
So,
k=shape parameter=4
θ=scale parameter=2
Hence, The shape and scale parameters have values of 4 and 2, respectively.
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public class BinarySearch \{ public static void main(Stringll args) f int [1]yl ist ={1,2,3,7,10,12,20}; int result = binarysearch ( inylist, 20); if (result =−1 ) System, out, println("Not found:"); else System.out.println("The index of the input key is " + result+ ". "): y public static int binarysearch(int]l List, int key) \{ int low =0; int high = iist. length −1 while (high >= low) \& int mid =( low + high )/2; if (key < List [mid] high = mid −1; else if (key =1 ist [ mid ] ) return inid; else low = mid +1; return −1; // Not found \} l TASK 4: Binary Search in descending order We have learned and practiced the implementation of the binary search approach that works on an array in ascending order. Now let's think about how to modify the above code to make it work on an array in descending order. Name your new binary search method as "binarysearch2". Implement your own code in Eclipse, and ensure it runs without errors. Submit your source code file (.java file) and your console output screenshot. Hint: In the ascending order case, our logic is as follows: int mid =( low + high )/2 if ( key < list [mid] ) else if (key = ist [mid]) return mid; In the descending order case; what should our logic be like? (Swap two lines in the above code.)
Answers
The task involves modifying the given code to implement binary search on an array in descending order. The logic of the code needs to be adjusted accordingly.
The task requires modifying the existing code to perform binary search on an array sorted in descending order. In the original code, the logic for the ascending order was based on comparing the key with the middle element of the list. However, in the descending order case, we need to adjust the logic.
To implement binary search on a descending array, we need to swap the order of the conditions in the code. Instead of checking if the key is less than the middle element, we need to check if the key is greater than the middle element. Similarly, the condition for equality also needs to be adjusted.
The modified code for binary search in descending order would look like this:
public static int binarysearch2(int[] list, int key) {
int low = 0;
int high = list.length - 1;
while (high >= low) {
int mid = (low + high) / 2;
if (key > list[mid])
high = mid - 1;
else if (key < list[mid])
low = mid + 1;
else
return mid;
}
return -1; // Not found
}
By swapping the conditions, we ensure that the algorithm correctly searches for the key in a descending ordered array.
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What are the zeros of the function? f(t)=t2−13t 36 Enter your answers in the boxes. And.
Answers
Answer:
t = 4, 9
Step-by-step explanation:
Hi there!
f(t) = t² - 13t + 36
Let f(t) = 0:
0 = t² - 13t + 36
Factor the equation:
0 = t² - 4t - 9t + 36
0 = t(t - 4) - 9(t - 4)
0 = (t - 9)(t - 4)
The zero-product property states that if the product of two numbers is zero, then one of the numbers is equal to zero. Therefore, either (t - 9) or (t - 4) is equal to zero:
t - 9 = 0
t = 9
t - 4 = 0
t = 4
Therefore, the zeros of the function are 4 and 9.
I hope this helps!