Determine whether the series is convergent. Identify the type of the series and test used to evaluate.
Answers
The first series,
\(\displaystyle \sum_{k=2}^\infty \frac{\cos(k)}{k^2}\)
is convergent. By comparison, using the fact that |cos(x)| ≤ 1 for all real x,
\(\displaystyle \sum_{k=2}^\infty \frac{\cos(k)}{k^2} \le \sum_{k=2}^\infty \frac1{k^2}\)
and the bounding series is a convergent p-series (with p = 2).
The second series,
\(\displaystyle \sum_{k=2}^\infty \frac{e^k}{\left(2+\frac1k\right)^k}\)
is divergent by the limit test. We have
\(\dfrac{e^k}{\left(2 + \frac1k\right)^k} = \dfrac{e^k}{2^k\left(1+\frac1{2k}\right)^k} = \left(\dfrac e2\right)^k \cdot \dfrac1{\left(1+\frac1{2k}\right)^k}\)
By definition,
\(e = \displaystyle \lim_{k\to\infty}\left(1+\frac1k\right)^k\)
so that
\(\displaystyle \lim_{k\to\infty}\left(1+\frac1{2k}\right)^k = \lim_{k\to\infty}\sqrt{\left(1+\frac1{2k}\right)^{2k}} = \sqrt{\lim_{k'\to\infty}\left(1+\frac1{k'}\right)^{k'}} = \sqrt{e}\)
so that the limit of the summand is
\(\displaystyle \lim_{k\to\infty} \frac{e^k}{\left(2+\frac1k\right)^k} = \frac1{\sqrt{e}} \lim_{k\to\infty} \left(\frac e2\right)^k\)
but e > 2, so the limit is ∞.
Related Questions
if m ∟n, solve for x. (x + 21) (3x + 9) giving brainliest
Answers
Answer:
x=15
Step-by-step explanation:
Add the two equations together because there is a 90 degree angle. You can add the two to get :
4x+30=90
-30 from both sides
4x=60
Divide by 4 on both sides to get your answer!
x=15
Answer: X = 15
I think you can give brainiest to the person above me now
I know it's really late but :)
e) Simplity
i. 10g3 27 ^3
Answers
Answer:
9
Step-by-step explanation:
Log₃27³
Recall that 27 = 3³
Hence
Log₃27³ = Log₃3³⁽³⁾ = Log₃3⁹
From the laws of logarithms, we know that
Logₐa = 1 and Log aˣ = xLog a
As such, applying both laws
Log₃3⁹ = 9Log₃3
since Log₃3 = 1
9Log₃3 = 9 * 1
= 9
Venetta makes an array with 5 rows of
6 columns. Jaime makes an array with
10 rows of 3 columns. Whose array
has more items? Explain your thinking.
Answers
Answer:
None/ Both
Step-by-step explanation:
to find the items in each array, multiply rows by columns. For Vendettas it would be 6x5, which is 30. For Jamie, it's 10x3 which is also 30. Because both arrays ended up with the same number, the answer is none/both.
Answer: both
Step-by-step explanation: since Venetta has an array of 5 rows and 6 columns, she has 5x6=30 items.
On the other hand, Jamie has 10 rows of 3 columns so she has 10x3=30 items.
Therefore, 30=30 so they both have the same amount of items in their arrays.
please i need HELPPP
Answers
Answer:
vertex point is (4,-6)
and it opens upwards as the coefficient of x is positive.
I assume B is the correct answer, good luck!
a)
percent.
A shopkeeper fixed the price of his articles 25 % above the cost price. If he sold an
article allowing 5% discount, find his profit percent.
Ashankeaner fived the more
bi
CL
Answers
60% of the books in a library are for adults, 5% are for young people and the rest are for children. If there are 280 books for children, how many books are there altogether?
Answers
Answer:
800 books
problem solving steps：
adults:60%
young people:5%
children=100%-60%-5%
=35%
35%=280 books
1%=280÷35
=8
100%=800
so,there are 800 books
What is (f−g)(x)? f(x)=3x5+6x2−5 g(x)=2x4+7x2−x+16
Answers
Answer: We have f(x)=3x⁵+6x²-5 and g(x)= 2x⁴+7x²-x+16
(f-g)(x)= 3x⁵-2x⁴-x²+x-21
Step-by-step explanation:
Here we have,
Given : f(x)=3x⁵+6x²-5 and g(x)= 2x⁴+7x²-x+16
We know,
(f-g)(x)= f(x)-g(x)
= (3x⁵+6x²-5 ) - ( 2x⁴+7x²-x+16)
On subtracting g(x) from f(x) we get,
(f-g)(x)= (3x⁵+6x²-5 - 2x⁴-7x²+x-16)
On simplify,
(f-g)(x) =3x⁵-2x⁴-x²+x-21
Hence,
(f-g)(x) = f(x) - g(x) = 3x⁵-2x⁴-x²+x-21
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Determine the length of side AB and BC in triangle ABC.
A. AB= 10.065, BC = 8.388
B. AB = 8.8, BC = 6.4
C. AB = 10.56, BC = 7.68
D. AB = 7.425, BC = 5.954
Answers
Answer:
C. AB=10.56, BC=7.68
Step-by-step explanation:
AB) 8.8x7.32÷6.1=10.56
BC) 6.4x7.32÷6.1=7.68
Which expression is equivalent to 1/4 -3/4 x?
Answers
Answer:
factor out 1/4 from the expression
1/4x(1-3x)
A sample of 311 people is selected. The people are classified according to place of residence ("urban", "suburban", or "rural"). They are also classified according to highest educational degree earned ("no college degree", "two-year degree", "four-year degree", or "advanced degree"). The results are given in the contingency table below. Urban Suburban Rural No college degree Two-year degree Four-year degree 34 42 22 35 21 22 16 34 16 Advanced degree 23 23 23 What is the relative frequency of people in the sample whose place of residence is suburban and whose highest degree is a two-year degree? Round your answer to two decimal places.
Answers
Answer: 0.07
Step-by-step explanation:
To find the relative frequency of people in the sample whose place of residence is suburban and whose highest degree is a two-year degree, we need to calculate the ratio of the number of people with those characteristics to the total sample size.
Looking at the contingency table, we can see that the number of people in the suburban category with a two-year degree is 21.
The total sample size is given as 311.
Therefore, the relative frequency can be calculated as:
Relative frequency = (Number of people in suburban category with two-year degree) / (Total sample size)
= 21 / 311
≈ 0.0676
Rounded to two decimal places, the relative frequency is approximately 0.07.
I need an answer to this asap, hope you can understand.
Answers
The lengths of the segments x and y in the right triangles are x = 6√2 and y = 12√2
How to determine the lengths of x and yThe given shape is the right-angled triangle
Such that
We have angles = 30 degrees and 45 degrees
The measure of y can be calculated using the following sine ratio
sin(45) = y/24
Make y the subject
So, we have
y = 24 * sin(45)
When evaluated, we have
y = 12√2
The length x is calculated as
sin(30) = x/y
So, we have
x = y * sin(30)
This gives
x = 12√2 * 1/2
Evaluate
x = 6√2
Hence, the value of x in the triangle is 6√2
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On Friday, a local hamburger shop sold a combined total of 396 hamburgers and cheeseburgers. The number of cheeseburgers sold was two times the number
of hamburgers sold. How many hamburgers were sold on Friday?
Answers
Answer:792
Step-by-step explanation:
An interior designer is redecorating a room that is 26 feet long by 18 feet wide by 9 feet high. At one end of the room is a door that is 6 feet 6 inches high and 4 feet wide. One of the walls contains 2 windows, each of which is 2 feet wide by 2 feet 6 inches high.
A: How much will it cost to carpet the floor if the carpet sells for $18.00 a square yard? $
B: How much will it cost to wallpaper all four walls if wallpaper costs $0.75 per square foot? $
C: How much will it cost to paint the ceiling using paint that sells for $25 per gallon if a quart of paint will cover 88 square feet? $
D: What will be the cost of the entire project? $
Answers
Answer: A: It will cost $936.00 to carpet the floor.
B: It will cost $297.00 to wallpaper all four walls.
C: It will cost $33.25 to paint the ceiling.
D: The cost of the entire project will be $1266.25.
Step-by-step explanation:
To calculate the costs for carpeting, wallpapering, painting, and the overall cost of the project, we need to determine the areas that need to be covered and the corresponding prices for each material.
Given dimensions:
Room length: 26 feet
Room width: 18 feet
Room height: 9 feet
Door dimensions:
Height: 6 feet 6 inches
Width: 4 feet
Window dimensions (each):
Width: 2 feet
Height: 2 feet 6 inches
A: Carpeting the floor:
To find the area of the floor, we multiply the length and width of the room:
Floor area = Length × Width = 26 feet × 18 feet = 468 square feet.
To convert to square yards (since the carpet is sold per square yard), we divide by 9:
Floor area in square yards = 468 square feet / 9 = 52 square yards.
Cost to carpet the floor = Floor area in square yards × Cost per square yard = 52 square yards × $18.00 = $936.00.
B: Wallpapering the walls:
To find the area of the walls, we calculate the perimeter of the room (2 × (Length + Width)) and multiply it by the height of the room:
Wall area = Perimeter × Height = 2 × (26 feet + 18 feet) × 9 feet = 396 square feet.
Cost to wallpaper the walls = Wall area × Cost per square foot = 396 square feet × $0.75 = $297.00.
C: Painting the ceiling:
To find the area of the ceiling, we multiply the length and width of the room:
Ceiling area = Length × Width = 26 feet × 18 feet = 468 square feet.
Since a quart of paint covers 88 square feet, we need to determine the number of quarts required:
Number of quarts = Ceiling area / Coverage per quart = 468 square feet / 88 square feet = 5.32 quarts.
Since a gallon contains 4 quarts, the number of gallons required is 5.32 quarts / 4 quarts = 1.33 gallons.
Cost to paint the ceiling = Number of gallons × Cost per gallon = 1.33 gallons × $25.00 = $33.25.
D: Cost of the entire project:
Total cost = Cost to carpet the floor + Cost to wallpaper the walls + Cost to paint the ceiling
= $936.00 + $297.00 + $33.25 = $1266.25.
Therefore:
A: It will cost $936.00 to carpet the floor.
B: It will cost $297.00 to wallpaper all four walls.
C: It will cost $33.25 to paint the ceiling.
D: The cost of the entire project will be $1266.25.
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Based on this theory, what distance will the handler move from the starting point to the return point if he creates an arc of a circle with radius 70 feet?
Group of answer choices
439.6 feet
3846.5 feet
109.9 feet
1758.4 feet
Answers
The Distance the handler will move from the starting point to the return point will be approximately 439.8204 feet.
The distance the handler will move from the starting point to the return point when creating an arc of a circle with a radius of 70 feet, we need to find the length of the arc.
The formula to calculate the length of an arc is given by:
Length of arc = (θ/360) * 2πr
Where:
θ is the central angle of the arc (in degrees)
r is the radius of the circle
In this case, since the handler is creating a full circle, the central angle is 360 degrees.
Length of arc = (360/360) * 2π * 70
Length of arc = (1) * 2π * 70
Length of arc = 2π * 70
Length of arc = 140π
To find the approximate value in feet, we can use the approximation π ≈ 3.14159.
Length of arc ≈ 140 * 3.14159
Length of arc ≈ 439.8204 feet
Therefore, based on the given theory and using a circle with a radius of 70 feet, the distance the handler will move from the starting point to the return point will be approximately 439.8204 feet.
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solve this question
Answers
Answer:
80
Step-by-step explanation:
Initial ratio:
Teachers to students
1 : 15
Ratio after:
Teachers to students
3 : 40
The number of students did not change, so we can make the students of the initial ratio and the students of the ratio after the same.
1 : 15 = 8 : 120
3 : 40 = 9 : 120
120 units = 1200
1 unit = \(\frac{1200}{120} =10\)
Initial teachers = 8 units = \(8*10=80\)
(02.04 LC)
Determine the equation of the line shown in the graph:
graph of a vertical line with an x intercept of negative 1
y = −1
y = 0
x = −1
x = 0
Answers
Answer:
x = - 1
Step-by-step explanation:
the equation of a vertical line is
x = c
where c is the value of the x- coordinates the line passes through
the line passes through the x- intercept of - 1 , then
x = - 1 ← equation of vertical line
Please help me 7th grade math
Answers
Answer: 1/36
Step-by-step explanation:
Your going to multiply 1/6 x 1/6 and you will get 1/36, because there are 6 slots, and you spin on x once, there is a 1/6 chance you will spin on it again, so that keeps a the 6/6 for all of the letters. But we are looking for f, there is also a 1/6 percent chance that it might land on that, 1/6 x 1/6 = 1/36
Answer:
Step-by-step explanation:
36 - 4
Step-by-step explanation:
(6x - 2)(6x + 2)
= (6x)(6x) + (6x)(-2) + (2)(6x) + (-2)(2)
= 36 - 12x + 12x - 4
= 36 - 4
Find the indefinite integral
Answers
Answer: \(\displaystyle \frac{2}{3}x^{3/2} + \frac{2}{5}x^{1/2}+C\\\\\\\)
This is equivalent to \(\frac{2}{3}\sqrt{x^3} + \frac{2}{5}\sqrt{x}+C\\\\\\\)
========================================================
Work Shown:
\(\displaystyle \int\left(\sqrt{x} + \frac{1}{5\sqrt{x}}\right)dx\\\\\\ \displaystyle \int\left(\sqrt{x}\right)dx + \int\left(\frac{1}{5\sqrt{x}}\right)dx\\\\\\ \displaystyle \int\left(x^{1/2}\right)dx + \int\left(\frac{1}{5}x^{-1/2}\right)dx\\\\\\ \displaystyle \int\left(x^{1/2}\right)dx + \frac{1}{5}\int\left(x^{-1/2}\right)dx\\\\\\\)
\(\displaystyle \frac{1}{1+1/2}x^{1+1/2} + \frac{1}{5}*\frac{1}{1+(-1/2)}x^{1+(-1/2)}+C\\\\\\ \displaystyle \frac{1}{3/2}x^{3/2} + \frac{1}{5}*\frac{1}{1/2}x^{1/2}+C\\\\\\ \displaystyle \frac{2}{3}x^{3/2} + \frac{1}{5}*2x^{1/2}+C\\\\\\ \displaystyle \frac{2}{3}x^{3/2} + \frac{2}{5}x^{1/2}+C\\\\\\\)
This graph shows a proportional relationship.
What is the constant of proportionality?
Answers
The constant of proportionality is a crucial aspect of a proportional relationship, as it tells us how much one variable changes in relation to the other. It can be calculated from the equation of the relationship, or from the slope of the graph if it is a linear relationship.
A proportional relationship is a mathematical equation that expresses two variables that are in proportion to one another. In this type of relationship, if one variable is multiplied by a certain constant factor, then the other variable will be multiplied by the same constant factor. For example, if the cost of 3 books is $15, then the cost of 6 books will be $30. In this case, the number of books and the cost of the books are in proportion to one another.
The constant of proportionality is the constant factor by which the two variables are multiplied to obtain each other. In other words, it is the ratio between the two variables that remain constant in a proportional relationship. It can be represented by the letter k, and is calculated by dividing one variable by the other.
To find the constant of proportionality from a graph, we need to look at the slope of the line. The slope is the ratio between the change in the y-values and the change in the x-values, which is also known as the rise over run. If the graph shows a proportional relationship, then the slope will remain constant throughout the graph.
For example, if the graph shows the relationship between the number of hours worked and the amount of money earned, and the slope is 10, then the constant of proportionality is 10. This means that for every hour worked, the person earns $10. The equation for this proportional relationship can be written as y = 10x, where y represents the amount of money earned and x represents the number of hours worked.
In conclusion, the constant of proportionality is a crucial aspect of a proportional relationship, as it tells us how much one variable changes in relation to the other. It can be calculated from the equation of the relationship, or from the slope of the graph if it is a linear relationship.
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Select the correct answer. 3 A linear function has a y-intercept of -12 and a slope of 3/2. What is the equation of the line?
a. y = 3/2x - 12
b.y=3/2x+12
c.y=2/3x-12
d y=-12x-3/2
Answers
Answer:
a. y = 3/2x - 12
Step-by-step explanation:
Slope is y = mx + b
Slope or m = 3/2
y-intercept or b = -12
Substitute m for 3/2 and b for -12 then the answer would be y = 3/2x - 12
Find the length of side x in simplest radical form
with a rational denominator.
30⁰
X
4
60°
Answers
The value of measure of x in the triangle is,
⇒ x = 4√3 units
We have to given that,
A triangle is shown in image.
Now, WE can formulate by trigonometry formula we get;
⇒ tan 30° = Opposite / Base
⇒ tan 30° = 4 / x
⇒ 1/√3 = 4/x
⇒ x = 4√3 units
Thus, The value of measure of x in the triangle is,
⇒ x = 4√3 units
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If -10 is subtracted from a number, the result is 6.
What is the number?
A. -16
B.-4
C. O
D. 4
Answers
Answer:
B.) -4
Step-by-step explanation:
-4-(-10)
-4+10
6
Which statement about the location of √7 on the number line is true?
A= It is located at the number 7 on the number line.
B= It is located at the number 3.5 on the number line.
C= It is located between the numbers 2 and 3 on the number line.
D=It is located between the numbers 4 and 9 on the number line
Answers
Explanation
The square root of a number is “what number times itself”
2 x 2 = 4
3 x 3 = 9
7 is between 4 and 9 so C is the correct answer
Which equation is equivalent to this equation and written with the same base?
4x+1=16x−1
Answers
Answer:
\( 2^{2x + 2} = 2^{4x - 4} \)
Step-by-step explanation:
\( 4^{x + 1} = 16^{x - 1} \)
\( 2^{2(x + 1)} = 2^{4(x - 1)} \)
\( 2^{2x + 2} = 2^{4x - 4} \)
The vectors v and w lie in the coordinate plane such that their initial points are at the origin. Vector v has a magnitude of 2 and direction of 45° North of East. Vector w has a magnitude of 2 and a direction of 45° South of East. What is the magnitude of the vector v+w?
Answers
The vector v+w has a magnitude of 2√2 and its direction is along the positive x-axis.
What is meant by vector?
A vector is a quantity that has both magnitude and direction. It is represented as an arrow with its length representing the magnitude and its direction representing the direction of the quantity.
What is meant by the x-axis?
The x-axis is the horizontal line on a coordinate plane that is used as a reference for plotting and describing the positions of points in two-dimensional space. It is often referred to as the "horizontal axis" or the "abscissa".
According to the given information
Here the vector v has a magnitude of 2 and a direction of 45° so the components of vector v are:
v_x = 2 cos 45° = √2
v_y = 2 sin 45° = √2
Vector w has a magnitude of 2 and a direction of 45° South of East. This means that the angle between vector w and the positive x-axis is 45°, and the angle between vector w and the negative y-axis is 45°. Therefore, the components of vector w are:
w_x = 2 cos 45° = √2
w_y = -2 sin 45° = -√2
Now we can add the components of vectors v and w to find the components of the vector v+w:
(v+w)_x = v_x + w_x = √2 + √2 = 2√2
(v+w)_y = v_y + w_y = √2 - √2 = 0
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5.7. Suppose n = 2911 and e = 11. Encrypt the following messages as in
Example (5.3).
a) "OK"
b) "HELP" (Break this up into two blocks.)
Answers
Note that,
the encrypted message for "OK" is the pair of numbers (616, 2385).
and the final encrypted message for "HELP" is the sequence of numbers (738, 1277, 1479, 2252).
To encrypt a message using RSA, we need to first represent the message as numbers using a suitable encoding scheme. For simplicity, we can use the ASCII code for each character, which is a standard encoding scheme for text.
a) To encrypt "OK", we first convert each letter to its corresponding ASCII code:
"O" = 79
"K" = 75
Next, we use the RSA encryption formula:
C ≡ \(M^{e}\) (mod n)
For "O", we have C ≡ 79¹¹ (mod 2911) ≡ 616 (mod 2911)
For "K", we have C ≡ 75¹¹ (mod 2911) ≡ 2385 (mod 2911)
Therefore, the encrypted message for "OK" is the pair of numbers (616, 2385).
b) To encrypt "HELP", we break it up into two blocks:
Block 1: "HE"
Block 2: "LP"
For block 1, we have:
"H" = 72
"E" = 69
Using the RSA encryption formula, we get:
C1 ≡ 72¹¹ (mod 2911) ≡ 738 (mod 2911)
C2 ≡ 69¹¹ (mod 2911) ≡ 1277 (mod 2911)
Therefore, the encrypted message for "HE" is the pair of numbers (738, 1277).
For block 2, we have:
"L" = 76
"P" = 80
Using the RSA encryption formula, we get:
C3 ≡ 76¹¹ (mod 2911) ≡ 1479 (mod 2911)
C4 ≡ 80¹¹ (mod 2911) ≡ 2252 (mod 2911)
Therefore, the encrypted message for "LP" is the pair of numbers (1479, 2252).
The final encrypted message for "HELP" is the sequence of numbers (738, 1277, 1479, 2252).
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A store owner buys sweatshirts for $16 and marks them up 20% to make a profit. The sales tax where his store is located is 7.5%. A customer considers buying one or two sweatshirts from the store. Select all the statements that apply.
One sweatshirt before tax costs $19.20.
One sweatshirt before tax costs $12.80.
The sales tax for two sweatshirts is $1.92.
The sales tax for two sweatshirts is $2.88.
Total cost for two sweatshirts is $41.28.
Total cost for two sweatshirts is $27.52.
Answers
Answer: The correct statements are:
1 SS before tax costs $19.20
Sales tax for 2 SS is $2.88
Total cost for 2 SS is $41.28
Step-by-step explanation:
$16 x 1.20 = $19.20 cost before tax
$19.20 x 0.075 = $1.44 sales tax, one shirt
Sales tax for 2 shirts = $1.44 x 2 = $2.88
Total cost of a shirt = $19.20 + $1.44 = $20.64
Total cost for 2 shirts = $20.64 x 2 = $41.28
Order from greatest to least -3, 6.5, 13 1/2, -5.3, 24, -1/4, and 1.25
Answers
Answer:
Step-by-step explanation:
Greatest to least would be 24, 13 1/2, 6.5, 1.25, -1/4, -3, -5.3
What is the Inverse Point of (1,5)?
Answers
Answer:
The Inverse point of (1,5)=(5,1)
Answer correct answer for brainliest
Answers
Answer:
Bro it’s a’c’d the sun the moon and the earth their energy works together pushing and pulling towards the earth creating a gravitational pull.
Step-by-step explanation:
Answer:
It's all of the above. Hope this helps :)
I may be wrong