Recall the Weierstrass product definition of the gamma function,
[tex]\displaystyle \frac1{\Gamma(z)} = ze^{\gamma z} \prod_{n=1}^\infty \left(1+\frac zn\right) e^{-z/n}[/tex]
from which we can obtain the log-gamma (i.e. logarithm of gamma) function,
[tex]\displaystyle -\ln\Gamma(z) = \ln(z) + \gamma z + \sum_{n=1}^\infty \left(\ln\left(1 + \frac zn\right) - \frac zn\right)[/tex]
where [tex]\gamma[/tex] is the Euler-Mascheroni constant.
Let [tex]z=1[/tex], so that
[tex]\displaystyle \gamma = \sum_{n=1}^\infty \left(\frac1n - \ln\left(1 + \frac 1n\right)\right)[/tex]
Recall the power series
[tex]\displaystyle \ln(1 + x) = - \sum_{n=1}^\infty \frac{(-x)^n}n[/tex]
Along with the definition of the Riemann zeta function,
[tex]\displaystyle \zeta(a) = \sum_{b=1}^\infty \frac1{b^a}[/tex]
we find that
[tex]\displaystyle \gamma = \sum_{n=1}^\infty \left(\frac1n + \sum_{m=1}^\infty \frac1m\left(-\frac1n\right)^m\right) \\\\ ~~~~ = \sum_{n=1}^\infty \sum_{m=2}^\infty \frac1m \left(-\frac1n\right)^m \\\\ ~~~~ = \sum_{m=2}^\infty \frac{(-1)^m}m \sum_{n=1}^\infty \frac1{n^m} \\\\ ~~~~ = \sum_{m=2}^\infty \frac{(-1)^m \zeta(m)}m[/tex]
so the original sum's value is [tex]\boxed{\gamma\approx0.577216}[/tex].
select two points that lie on the same line throught (0,0), then select an equation that describes the relationship between the coordinates x and y
Two collinear points on a line are given in the table below:
x, y
0 0 (0,0)
2 1 (2,1)
Select the points that lie on the line. and the relationship between the coordinates x and y.
(4,2)
(4, 3)
(7,2)
4/8, 2/8
Answer:
(4,2) and ([tex]\frac{4}{8}[/tex],[tex]\frac{2}{8}[/tex]) are lie on the line, and y = [tex]\frac{1}{2}[/tex] x , is showing the relationship of x and y coordinates of given line.Given that
The co-ordinates of line is (0,0) and (2,1).To find
The points that lie on the line, and The relationship between the coordinates x and y.So, according to the question
We have,
The X and Y co-ordinates of line (0,0) and (2,1).For finding any point that lie on line, first we have to find the slope of given line,
So, we know that
slope (m) = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
Where,
x₁ = 0 and x₂ = 2
y₁ = 0 and y₂ = 1
Now, putting the values, we will get
(m) = [tex]\frac{1-0}{2-0}[/tex] = [tex]\frac{1}{2}[/tex]
m = [tex]\frac{1}{2}[/tex]
We know that, the equation of line
y-y₁ = m(x- x₁ )
y-0 = [tex]\frac{1}{2}[/tex] (x- 0)
y = [tex]\frac{1}{2}[/tex] x -------------(1)
{ this equation (1) is showing a relationship the coordinates x and y }
For finding which point is lie in that line, this point must be prove that equation (1).
Now, let's check the points
For point (4,2)
y = [tex]\frac{1}{2}[/tex] x -------------(1)
2 = [tex]\frac{1}{2}[/tex] × 4
2 = 2
So, point (4,2) lie in this line.
For point (4,3)
y = [tex]\frac{1}{2}[/tex] x -------------(1)
3 = [tex]\frac{1}{2}[/tex] × 4
3 ≠ 2
So, point (4,3) not lie in this line.
For point (7,2)
y = [tex]\frac{1}{2}[/tex] x -------------(1)
2 = [tex]\frac{1}{2}[/tex] × 7
2 ≠ [tex]\frac{7}{2}[/tex]
So, point (7,2) not lie in this line.
For point ([tex]\frac{4}{8}[/tex],[tex]\frac{2}{8}[/tex])
y = [tex]\frac{1}{2}[/tex] x -------------(1)
[tex]\frac{2}{8}[/tex] = [tex]\frac{1}{2}[/tex] × [tex]\frac{4}{8}[/tex]
[tex]\frac{2}{8}[/tex] = [tex]\frac{2}{8}[/tex]
So, point ([tex]\frac{4}{8}[/tex],[tex]\frac{2}{8}[/tex]) lie in this line.
(4,2) and ([tex]\frac{4}{8}[/tex],[tex]\frac{2}{8}[/tex]) are lie on the line, and y = [tex]\frac{1}{2}[/tex] x , is showing the relationship of x and y coordinates of given line.
To learn more about line, please click on the link:
https://brainly.com/question/7100287
#SPJ9
What is the Area of the Figure?
Answer: 58 sq ft
Step-by-step explanation:
2 * 2 = 4
9 * 6 = 54
54+4=58
3 Choose 1 answer: A B D x = √√√27 x = √18 x = 5 x = 4 6
Answer:
the answer is A-x=
Step-by-step explanation:
because if you multiply 3 by 6 you get 27
4. Old Faithful, Yellowstone National Park's most famous
geyser, shoots a spray of steam and hot water 50 m
into the air. Is this height greater or less than 5,000 mm?
Greater because a meter is greater than a millimeter.
Also can I get the brainliest pls?
A toilet manufacturer has decided to come out with a new and improved toilet. The fixed cost for the production of this new toilet line is $16,600 and the variable costs are $67 per toilet. The company expects to sell the toilets for $157. Formulate a function P(x) for the total profit from the production and sale of x toilets.
The function P(x) for the total profit from the production and sale of x toilets is P(x) = 90x - 16600
How to formulate a function P(x) for the total profit from the production and sale of x toilets?The given parameters are
Fixed cost = 16600
Variable cost = 67 per toilet
Cost of toilet = 157 per toilet
Let the number of toilets be x
So, the cost function is
C(x) = Fixed cost + Variable cost * Number of toilets
This gives
C(x) = 16600 + 67x
Also, we have the revenue function to be
R(x) = Cost of toilet * Number of toilets
This gives
R(x) = 157x
The function P(x) for the total profit from the production and sale of x toilets is
P(x) = R(x) - C(x)
So, we have
P(x) = 157x - 16600 - 67x
Evaluate
P(x) = 90x - 16600
Hence, the function P(x) for the total profit from the production and sale of x toilets is P(x) = 90x - 16600
Read more about profit at
https://brainly.com/question/19104371
#SPJ1
identify the coefficient, constant, and variable in the expression: - 5 + 8w
Answer:
coefficient=8
constant=-5
variable=x
Step-by-step explanation:
Answer:
-5 constant
8 coefficient
w variable
Step-by-step explanation:
What is the greatest common factor of 38 and 30?
The greatest common factor of 38 and 30 is 2.
How to calculate the value?It should be noted that thw first step to this method of finding the Greatest Common Factor of 30 and 38 is to find and list all the factors of each number.
Let’s take a look at the factors for each of these numbers, 30 and 38:
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
Factors of 38: 1, 2, 19, 38
When you compare the two lists of factors, you can see that the common factor(s) are 1, 2. Since 2 is the largest of these common factors, the GCF of 30 and 38 would be 2.
Learn more about factor on:
https://brainly.com/question/25829061
#SPJ1
Robert made 3 more than twice as many baskets as Billy did in the
basketball game.
The algebraic expressions of the statements are x = 6y - 4 and r = 3 + 2b
How to determine the statement as an algebraic expression?Statement 1
The statement is given as
Ramona has 4 less than 6 times as many one dollar bills as Rita has
Let the number of one dollar bills owned by Ramona be x and the number of one dollar bills made by Rita be y
So, the interpretation of the statement is
x = 6y - 4
Hence, the algebraic expression of the statement is x = 6y - 4
Statement 2
The statement is given as
Robert made 3 more than twice as many baskets as Billy did in the basketball game.
Let the number of baskets made by Robert be r and the number of baskets made by billy be b
So, the interpretation of the statement is
r = 3 + 2b
Hence, the algebraic expression of the statement is r = 3 + 2b
Read more about expressions at
https://brainly.com/question/723406
#SPJ1
Which expression shows the first step in dividing 5x + 3 by x?
The First step Linear Expression shows is 5+3/x
An algebraic expression known as a linear expression has terms that are either constants or variables raised to the first power. Alternatively put, none of the exponents can be greater than 1. As an illustration, while x is a variable raised to the first power, x2 is a variable raised to the second power. An illustration of a constant is 5.
so when this linear expression 5x + 3 is divided by x then the first step will be
5 + 3/x
Read more about linear expression here:
https://brainly.com/question/819568
#SPJ4
Solve
4x + 8y = 16 for X.
Solve for x
Answer:
x = 4 - 2y
Step-by-step explanation:
4x + 8y = 16
or, x + 2y = 4
or, x = 4 - 2y
Answer:4-2y
Step-by-step explanation:
Is 7.65 greater or less than 7.7
Answer:
less than
Step-by-step explanation:
is this proportion or not a meal delivery service cost $10 to sign up ,plus $24.99 each month ASAP
Answer:
This is not proportional.
Step-by-step explanation:
If there was no meal delivery cost than it would be proportional.
A superhero recently asked his nemesis how many cats she has. She answered with a riddle: "One-half of my cats plus seven." How many cats does the nemesis have?
Answer:
14
Step-by-step explanation:
You want the number of my cats when that number is "one-half of my cats plus seven".
SetupLet the number of cats be represented by c. Then the description of the number of cats is ...
c = c/2 +7
SolutionMultiplying by 2 gives ...
2c = c +14
c = 14 . . . . . . . subtract c
The nemesis has 14 cats.
what is the solution to -3/8x-11=4
Answer: -40
Step-by-step explanation:
Answer:
-1/40
Step-by-step explanation:
-3/8x -11+11=4+11
-3/8x=15
-3/8x x=15x
-3/8=15x
15x/15= -3/8/15
x=-1/40
Hope this helps!^^
If the difference of a number and eight is tripled, the result is six more than the number. Find the number. Round your answer to the nearest integer, if necessary.
The number that is six less than the product of three and the difference of the number and eight is; 15
How can the required number in the word problem be found?The verbal expression of the equation in the word problem is presented in mathematical form as follows;
(x - 8) × 3 = x + 6Expanding the left hand side gives;
3•x - 3×8 = x + 6
3•x - 24 = x + 6
3•x - x = 24 + 6 = 30
2•x = 30
x = 30/2 = 15
The number, x = 15Learn more about finding the solutions to word problems here:
https://brainly.com/question/13818690
#SPJ1
NEED ANSWERS ASAP WE NEED IT TOMORROW GRADE SIX MATH
1.What is the difference when ⅓ is subtracted from ¾?
2.The sum of ³/⁷ and ½ is equal to? ___
3.The difference between ⅘ and ¼ is ___
4. The sum of 2¼ and 1³/²⁴ subtracted by 1¾ is ______
write one hundred sixty-six thousandths in standar form
0.166 is the standard form of that
What are the next two terms in the following sequence? 6, 3, 12, 6, 24.....
Answer:
12, 48
The pattern is: Divide by 2, then multiply by 4
greg sets his watch for the correct time exactly one week later he finds that his watch 1 1 over 3 minutes it loses time at the same rate for eight weeks write a multiplication expression that describes the sittuation find the product and explain how it is related to the problem?
The product or overall change in time for 8 weeks will be 29.33 minutes.
What is a statement?A statement is a declarative sentence that is either true or false but not both. A statement is sometimes called a proposition. The key is that there must be no ambiguity. To be a statement, a sentence must be true or false, and it cannot be both.
Given:
After one week, Greg's watch has lost 11/3 minutes. If the watch continues to lose time at the same rate, we can say that the change in time is 11/3 minutes per week.
According to given question we have
(11/3 minutes/week) x (8 weeks) = 29.33 minutes
Therefore, the product or overall change in time for 8 weeks will be 29.33 minutes.
Learn more details about statement here:
https://brainly.com/question/2285414
#SPJ1
Can somebody please help me I'm trying to find out what 2 + 2 is
Answer:
2+2 = 4
Step-by-step explanation:
Let's visualize this using # signs
we have two groups of two # signs, so let's map that out
##
##
If you count the individual # signs, it is 4. You can also use this model to figure out 2 *2 (4)
I hope this helps :)
Answer:
4
Step-by-step explanation:
2+2
2+1+1
3+1
4
Being... Great.... might be 78?! nah she prolly not im jk
Answer:
B.
Step-by-step explanation:
233% = 2.33
2 2/3 = 2.6
All work and the answer is provided in the screenshot that is attached! :)
Have a great day!
On Monday, Saffron bakes
324 pastries in the morning and some more in the
afternoon. She sells 584 pastries. At the end of the
day she has 52 pastries left. How many pastries did
she bake in the afternoon?
the number of pastries that she bake in the afternoon is equal to 312 pastries.
We are given that:
Saffron sells 584 pastries in the whole day and 52 pastries are left at the end of the day.
Now, we will use addition to find the total number of pastries that is being made in the day.
Total pastries = 584 + 52
Total pastries = 636
Now, she bake 324 pastries in the morning.
So, the number of pastries baked in the afternoon = 636 - 324
In afternoon = 312.
Therefore, we get that the number of pastries that she bake in the afternoon is equal to 312 pastries.
Learn more about addition here:
https://brainly.com/question/25421984
#SPJ9
Can someone help me get the points I’ll make u brilliant the points are S T R U
Using translation concepts, the coordinates of the vertices after a rotation 180º clockwise around the origin is:
S': (-4,-10), R'(-4,-3), U'(-8,-3) and T'(-8,-10).
What is a translation?A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction either in it’s definition or in it’s domain. Examples are shift left/right or bottom/up, vertical or horizontal stretching or compression, and reflections over the x-axis or the y-axis.
For this problem, the coordinates (x,y) of the vertices are given as follows:
S(4,10), R(4,3), U(8,3), T(8,10).
The rule for a rotation 180º clockwise around the origin is:
(x,y) -> (-x,-y).
Hence the vertices of the rotated image will be given as follows, applying the above rule:
S': (-4,-10), R'(-4,-3), U'(-8,-3) and T'(-8,-10).
More can be learned about translation concepts at https://brainly.com/question/4521517
#SPJ1
If an arrow is shot upward on the moon with a velocity of 58 m/s, its height in meters after t seconds is given by y=58t−.83t2.
(a) Find the average velocity over the given time intervals:
(1) [1,1.5]
(2) [1,1.1]
(3) [1,1.01]
(4) [1,1.001]
(b) Find the instantaneous velocity after one second (to the nearest hundredth).
(1) m/s
(2) m/s
(3) m/s
(4) m/s
(b) m/s
The 58 m/s initial velocity of the arrow with height, y = 58•t - 0.83•t², gives;
(1) 55.925 m/s
(2) 56.257 m/s
(3) 56.3317 m/s
(4) 56.33917 m/s
(b) 56.34 m/s
How can the average and instantaneous velocity be calculated?Initial velocity of the arrow = 58 m/s
Height of the arrow is; y = 58•t - 0.83•t²
The above kinematic equation can be analyzed as follows;
(1) The height at t = 1 is found as follows;
y = 58 × 1 - 0.83 × 1² = 57.17
Height at t = 1.5 s. is y = 58×1.5 - 0.83×1.5² = 85.1325
Average velocity = ∆y/∆tWhich gives;
Average velocity = (85.1325-57.17)/(1.5-1) = 55.925
The average velocity in the interval [1, 1.5] is therefore;
55.925 m/s(2) [1, 1.1]
Height at t = 1.1 s. is y = 58×1.1 - 0.83×1.1² = 62.7957
Which gives;
Average velocity = (62.7957-57.17)/(1.1-1) = 56.257
The average velocity is 56.257 m/s(3) [1, 1.01]
Height at t = 1.01 s. is y = 58×1.01 - 0.83×1.01² = 57.733317
Which gives;
Average velocity = (57.733317-57.17)/(1.01-1) = 56.3317
The average velocity is 56.33 m/s
(4) [1, 1.001]
Height at t = 1.01 s. is y = 58×1.001 - 0.83×1.001² = 57.22633917
Which gives;
Average velocity = (57.22633917-57.17)/(1.001-1) = 56.33917
The average velocity is 56.33917 m/s(b) The equation for the instantaneous velocity is found as follows;
v = dy/dt = 58 - 2×0.83•t
After 1 second, we have;
t = 1
v = 58 - 2×0.83 × 1 = 56.34
The instantaneous velocity at t = 1 seconds is 56.34 m/s
Learn more about kinematics here:
https://brainly.com/question/14511323
#SPJ1
HELP ME OUT PLEASE!!!!!!!!
Answer:
-3
Step-by-step explanation:
Answer:
x-intercepts: (6,0)
y-intercepts: (0,3)
A square acre of land is 5,280 square yards.
Between which two integers is the length of one side?
between 17 and 18 yards
between 72 and 75 yards
between 24 and 243 yards
between 1,320 and 1,321 yards
By factor decomposition, area of a square and power and root properties, the length of the side is between 72 yards and 76 yards. (Correct choice: B)
What integers are the limits between the length of an equivalent square side of a given area?
The area of a triangle is equal to the square of its side length and the area of land is a composite number, that is, a product of prime numbers. By factor decomposition, the equivalent form of 5,280 square yards is:
5,280 = 2⁵ × 3 × 5 × 11
Then, the equivalent side is equal to the square product of the product:
x = √(2⁵ × 3 × 5 × 11)
x = √(2⁴) × √(2 × 3 × 5 × 11)
x = 2² × √(2 × 3 × 5 × 11)
x = 4√330
As for √330 is between √324 = 18 and √361 = 19, then the length of the side is between 72 yards and 76 yards.
To learn more on factor decomposition: https://brainly.com/question/13175237
#SPJ1
1/3 divided by 3/5 = 1/3 divided by M. What is M. I came up with 3/5
Answer:
Yes you are correct
Step-by-step explanation:
1/3 / 3/5 = 1/3 / M
3*1/3 / 5*3/5 = 3*1/3 / 5M
3/3 / 15/5 = 3/3 / 5M
1/3=1/(5M)
5M * 1/3 = 1
5M * 3 * 1/3 = 1 * 3
5M=3
M=3/5 √
Doug says the area of the figure is 20 square units. What is the correct area of
the figure? What mistake might Doug have made?
5 units
5 units
Answer:
The correct area is 25 square units.
His mistake was that he found the perimeter instead of the area.
Step-by-step explanation:
Area = Length x Width
Area= 5 x 5= 25
His mistake:
Perimeter = length + length + width + width
Perimeter = 5 + 5+ 5+ 5 = 20
Factor out the greatest common factor.
3x³-18x² + 12x
Answer:
x² - 6x + 4
Step-by-step explanation:
Hello!
Factor each term completely:
3x³ = (3)(x)(x)(x)-18x² = (-1)(2)(3)(3)(x)(x)12x = (2)(2)(3)(x)Common factors that are shown in all 3 terms are x and 3.
Divide each term by 3x:
(3)(x)(x)(x) ÷ 3x = (x)(x) = x²(-1)(2)(3)(3)(x)(x) ÷ 3x = (-1)(2)(3)(x) = -6x(2)(2)(3)(x) ÷ 3x = (2)(2) = 4Combine the terms: x² - 6x + 4
someone please help me out with this need for important grade
Answer:
6
Step-by-step explanation:
1605 - 75
1530 / 255 = 6