Let [tex]S[/tex] be the sum. Introduce a factor of [tex]\Gamma\left(\frac12\right)=\sqrt\pi[/tex] to rewrite the summand as a beta function integral. Then in the sum, interchange it with the integral, and [tex]S[/tex] reduces to a simple integral.
[tex]\displaystyle S = \sum_{n=0}^\infty \left(-\frac12\right)^n \frac{\Gamma\left(\frac{n+1}2\right)}{\Gamma\left(\frac n2+1\right)}[/tex]
[tex]\displaystyle . ~~ = \frac1{\sqrt\pi} \sum_{n=0}^\infty \left(-\frac12\right)^n \, \mathrm{B}\left(\frac n2+\frac12, \frac12\right) \\\\ ~~~~ = \frac1{\sqrt\pi} \sum_{n=0}^\infty \left(-\frac12\right)^n \int_0^1 \frac{t^{n/2}}{\sqrt t \sqrt{1-t}} \, dt \\\\ ~~~~ = \frac1{\sqrt\pi} \int_0^1 \frac{dt}{\sqrt t \sqrt{1-t}} \sum_{n=0}^\infty \left(-\frac{\sqrt t}2\right)^n \\\\ ~~~~ = \frac2{\sqrt\pi} \int_0^1 \frac{dt}{\sqrt t\sqrt{1-t}\left(\sqrt t+2\right)}[/tex]
Substitute [tex]u=\sqrt t+2[/tex] and [tex]du=\frac{dt}{2\sqrt t}[/tex].
[tex]\displaystyle S = \frac4{\sqrt\pi} \int_2^3 \frac{du}{u\sqrt{-(u-1)(u-3)}}[/tex]
Now substitute [tex]u=1+\frac2{1+t^2}[/tex] and [tex]du=-\frac{4t}{(1+t^2)^2}\,dt[/tex] - this comes from an Euler substitution of the form [tex]\sqrt{a(x-\alpha)(x-\beta)}=(x-\alpha)t[/tex]. [tex]S[/tex] reduces drastically to a trivial arctangent integral.
[tex]\displaystyle S = \frac8{\sqrt\pi} \int_0^1 \frac{dt}{t^2+3} \\\\ ~~~~ = \frac8{\sqrt\pi} \cdot \frac\pi{6\sqrt3} = \boxed{\frac43\sqrt{\frac\pi3}}[/tex]
I need help to find x and y
The system of nonlinear equations have two solutions: (x, y) = (0.114, 0.354), (x, y) = (1.432, 1.103).
What is the solution of a system of nonlinear equations?In this problem we find a system of two nonlinear equations involving radical expressions. We proceed to resolve the system of nonlinear equations by algebra properties:
(2 · x - 1)² + 8√(2 · x · y) = 4
8√(2 · x · y) = 4 - (2 · x - 1)²
8√(2 · x · y) = 4 - 4 · x² + 4 · x - 1
8√(2 · x · y) = 4 · x - 4 · x² + 3
64 · (2 · x · y) = [(4 · x + 3) - 4 · x²]²
128 · x · y = (4 · x + 3)² - 8 · (4 · x + 3) · x² + 64 · x⁴
128 · x · y = 16 · x² + 24 · x + 9 - 32 · x³ - 24 · x² + 64 · x⁴ (1)
4 · y - √(8 · x · y - 1) = 1
4 · y - 1 = √(8 · x · y - 1)
(4 · y - 1)² = 8 · x · y - 1
16 · y² - 8 · y + 2 = 8 · x · y (2)
Now we plot two equations on a graphing tool and find that the system has the following two solutions: (x, y) = (0.114, 0.354), (x, y) = (1.432, 1.103).
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One state has a 6% sales tax on clothing items priced at $75 or higher, and no sales tax on clothing items priced under $75. What is the total tax on the items in the table above?
two complementary angles are X + 4 degree and 2 x - 7 degree find the value of x
Answer:
x+4+2x-7=180
3x-3=180
3x=183
x=61
Given the graphs of f(x) and g(z) below, find the composition of functions f(g(-1)).
The composition of the functions, f(g(-1)) = 2.
How to Find the Composition of Functions?To find the composition of the functions, f(g(-1)), first, find the function, g(-1) by tracing the value of y that will give an x-value of -1 on the function graph.
The next step is to use the value you get in the first step to trace which value of y will give an x-value equivalent to what you got in the first step on the second function graph.
Thus, from the graph of g(x), g(-1) = 0. Using the graph of f(x), the value of y when x = 0 is 2.
Therefore, the composition of the functions, f(g(-1)) = 2.
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finding the common difference
5,11,____ , 23,29,____…
Answer:
5,11 , 17, 23, 29, 35
Step-by-step explanation:
Answer:
5, 11, 17, 23, 29, 36
Step-by-step explanation:
To get from 5 to 11, we add 6. Similarly, to get from 11 to _, we add 6, and get 17. This also works with 29 to _. We get 36.
Make a conjecture about each pattern. Then write the next two items. 4. 1, 2, 2, 4, 8, 32, . . .
Answer:
256, 8192
{This is an educated guess}
Step-by-step explanation:
The problem seems to be a variation of the Fibonacci sequence: 1, 1, 2, 3, 5, 8...
1 ⇒ 2: 1*2
2 ⇒ 2: 2*1
2 ⇒ 4: 2*2
4⇒8:4*2
8⇒32:8*4
32⇒256: 32*8
256*32 = 8192
Express 88 kilometers per hour in miles per hour.
mi/hr
(Round to the nearest hundredth as needed.)
88 kilometers per hour in miles per hour is 54.68 mi/hr
Converting kilometers per hour to miles per hourNote that:
1 km = 0.6214 miles
The measurement to convert is 88 kilometers per hour
Multiply 88 kilometers per hour by 0.6214 to convert to miles per hour
88 kilometers per hour = 88 x 0.6214 miles per hour
88 kilometers per hour = 54.6832 miles per hour
88 kilometers per hour = 54.68 mi/hr
Therefore, 88 kilometers per hour in miles per hour is 54.68 mi/hr
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Solve.
3z +2y-z = 8
2z+22=-4
z+3y=4
Answer:
(1, 1,-3)
Step-by-step explanation:
For the piecewise function, find the values g(-6), g(1), and g(8).
The output values of g(-6), g(1), and g(8) in the given piecewise function are -1, 6 and -6 respectively.
What are the output values of g(-6), g(1), and g(8) in the given piecewise function?Given the piecewise function in the question;
x + 5, for x ≤ 1
g(x) = {2 - x, for x > 1
g(-6) = ?g(1) = ?g(8) = ?Determine the output value of g(-6), -6 falls in the domain of x ≤ 1,
Hence;
g(x) = x + 5
g(-6) = -6 + 5
g(-6) = -1
Determine the output value of g(1), 1 falls in the domain of x ≤ 1,
Hence;
g(x) = x + 5
g(1) = 1 + 5
g(1) = 6
Determine the output value of g(8), 8 falls in the domain of x > 1,
Hence;
g(x) = 2 - x
g(8) = 2 - 8
g(8) = -6
Therefore the output values of g(-6), g(1), and g(8) in the given piecewise function are -1, 6 and -6 respectively.
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Tom bought a pencil case for $60 and sold it to gain a profit of 20% on his cost price.
(a) How much money did he gain? (b) How much money did he sell the pencil case for?
For the function
f(x) = x + 10/6x + 4, consider the following.
(a)Find the vertical and horizontal asymptotes for the graph of f.
Vertical:
Horizontal:
b)Find f^ −1.
f^ −1(x) =
c)Find the vertical and horizontal asymptotes for the graph of
f^ −1
Vertical:
Horizontal:
Show all steps.
Using the concepts of asymptotes and inverse function, we have that:
a) For f, the vertical asymptote is x = -2/3 and the horizontal asymptote is y = 1/6.
b) The inverse function is: [tex]y = f^{-1}(x) = \frac{4x - 10}{1 - 6x}[/tex]
c) For the inverse function, the vertical asymptote is x = 1/6 and the horizontal asymptote is y = -2/3.
What are the asymptotes of a function f(x)?The vertical asymptotes are the values of x which are outside the domain, which in a fraction are the zeroes of the denominator.The horizontal asymptote is the value of f(x) as x goes to infinity, as long as this value is different of infinity.For this problem, the function is given by:
[tex]f(x) = \frac{x + 10}{6x + 4}[/tex]
The zero of the denominator is:
6x + 4 = 0
6x = -4
x = -4/6 = -2/3.
Hence the vertical asymptote is x = -2/3.
The horizontal asymptote is given as follows:
[tex]y = \lim_{x \rightarrow \infty} f(x) = \lim_{x \rightarrow \infty} \frac{x + 10}{6x + 4} = \lim_{x \rightarrow \infty} \frac{x}{6x} = \lim_{x \rightarrow \infty} \frac{1}{6} = \frac{1}{6}[/tex]
How to find the inverse function?To find the inverse function of a function y = f(x), we exchange x and y then isolate y.
Exchanging x and y, we have that:
[tex]x = \frac{y + 10}{6y + 4}[/tex]
Applying cross multiplication:
x(6y + 4) = y + 10
6xy - y = 10 - 4x
y - 6xy = 4x - 10
y(1 - 6x) = 4x - 10
[tex]y = f^{-1}(x) = \frac{4x - 10}{1 - 6x}[/tex]
Applying the same procedure, for the inverse function, the vertical asymptote is x = 1/6 and the horizontal asymptote is y = -2/3.
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One side of a rectangle is 4 m longer than four times another side. The area of the rectangle is 224 m².
Find the length of the shorter side.
m
Let the another side be x m.
Therefore, the longer side = (4 + 4x) m.
Now,
[tex]{\sf{→x \times (4 + 4x) = 224 \\ →4x + 4x {}^{2} - 224 = 0 \\→ x {}^{2} + x - 56 = 0 \\ →x {}^{2} + 8x - 7x - 56 = 0 \\ →x(x + 8) - 7(x + 8) = 0 \\ →(x - 7)(x +8 ) = 0 \\ →x = 7 \: or \: x = - 8 \\ \\ length \: cannot \: be \: negative \\ possible \: solution \: (x) = 7}}[/tex]
Length of shorter side = x = 7 metres.
[tex]{\sf{answer\: \\ss}}[/tex]
EASY QUESTION 20 POINTS
Answer:(7/9)^12
Step-by-step explanation:
Answer:
C. (7/9)^12
Step-by-step explanation:
(49/81)^6 = (7^2/9^2)^6 = (7/9)^(2x6) = (7/9)^12
what is the domain and range of the question below
domain=
range =
The Domain is [-4, -2] U [-1, 1) and range is [4, 0)
What is Domain and Range?The domain of a function is the set of values that we are allowed to plug into our function.
The range of a function is the set of values that the function assumes.
Given:
The left piece has its other endpoint at x = -2.
So, the interval for the domain is [-4, -2] and the other part is [-1, 1)
So, the domain is [-4, -2] U [-1, 1)
and the range is the vertical range of the right part.
So, range is [4, 0)
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Graph the line with a slope of 1/3 passing through point (-1, -1)
Considering the expression of a line, the line is y= 1/3x -2/3 and the graph of this line is shown in the attached image.
Definition of lineA linear equation o line can be expressed in the form y = mx + b
where
x and y are coordinates of a point.m is the slope.b is the ordinate to the origin and represents the coordinate of the point where the line crosses the y axis.Knowing the value of slope m, substituting this value and the value of the point in the expression of a linear equation, y = mx + b, the value of the ordinate to the origin b can be obtained.
Line in this caseIn this case, you know:
The line has a slope of 1/3The line passes through point (x,y)=(-1, -1)Substituting the value of the slope m:
y= 1/3x +b
Substituting the point to calculate the value of b:
-1= 1/3×(-1) + b
-1= -1/3 + b
-1 + 1/3= b
-2/3= b
Finally, the line is y= 1/3x -2/3 and the graph of this line is shown in the attached image.
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[tex] \rm \int_{0}^1 \int_{0}^1x \bigg \{ \frac{1}{1 - xy} \bigg \}dydx \\ [/tex]
The fractional part vanishes when the argument is an integer; in this case, for
[tex]\left\{\dfrac1{1-xy}\right\} = 0 \iff \dfrac1{1-xy} = n \iff xy = \dfrac{n-1}n[/tex]
which are hyperbolas in the [tex](x,y)[/tex]-plane.
Observe that between neighboring hyperbolas, we have
[tex]\dfrac{n-1}n < xy < \dfrac n{n+1} \\\\ ~~~~ \implies \dfrac1{n+1} < 1-xy < \dfrac1n \\\\ ~~~~ \implies n < \dfrac1{1-xy} < n+1 \\\\ ~~~~ \implies \left\{\dfrac1{1-xy}\right\} = \dfrac1{1-xy} - \left\lfloor\dfrac1{1-xy}\right\rfloor = \dfrac1{1-xy} - n[/tex]
Split up the integral over [tex][0,1)^2[/tex] along the curves [tex]xy=\frac{n-1}n[/tex]. The subregions somewhat resemble the layers or scales of an onion (see attached plot with the first 5 "scales").
Let [tex]S_n[/tex] denote the [tex]n[/tex]-th ([tex]n\in\Bbb N[/tex]) "scale", starting from the blue region closest to the origin and counting diagonally upward in the direction of (1, 1).
In Cartesian coordinates, the integral over [tex]n[/tex]-th "scale" is
[tex]\displaystyle \iint_{S_n} x \left(\frac1{1-xy} - n\right) \, dy \, dx \\\\\\ ~~~~~~~~= \int_{(n-1)/n}^{n/(n+1)} \int_{(n-1)/(nx)}^1 x \left(\frac1{1-xy} - n\right) \, dy dx \\\\\\ ~~~~~~~~~~~~~ + \int_{n/(n+1)}^1 \int_{(n-1)/(nx)}^{n/((n+1)x)} x \left(\frac1{1-xy} - n\right) \, dx[/tex]
(see attached plot of the 2nd "scale" for reference)
The integral is trivial, so I'll leave it to you to confirm that it drastically reduces to
[tex]\displaystyle \iint_{S_n} x \left(\frac1{1-xy} - n\right) \, dy \, dx = \frac1{2n (n+1)^2} = \frac12 \left(\frac1n - \frac1{n+1} - \frac1{(n+1)^2}\right)[/tex]
Now we recover the original integral by summing over [tex]\Bbb N[/tex].
[tex]\displaystyle \int_0^1 \int_0^1 x \left\{\frac1{1-xy}\right\} \, dy \, dx = \frac12 \sum_{n=1}^\infty \left(\frac1n - \frac1{n+1} - \frac1{(n+1)^2}\right) \\\\ ~~~~~~~~ = \frac12 \left(\left(1-\frac12\right)+\left(\frac12-\frac13\right)+\left(\frac13-\frac14\right)+\cdots\right) - \frac12 \sum_{n=2}^\infty \frac1{n^2} \\\\ ~~~~~~~~ = \frac12 - \frac12 \left(\sum_{n=1}^\infty \frac1{n^2} - 1\right) \\\\ ~~~~~~~~ = \frac12 - \frac12 \left(\frac{\pi^2}6 - 1\right) = \boxed{1 - \frac{\pi^2}{12}}[/tex]
20 points!!!What is a two step problem with variables on each side equal 20? (Please help)
5x-15=5+4x
+15 +5
5x=20+4x
-4x -4x
x= 20
Step-by-step explanation:
Calculate (multiply (1-2i)(6+5i)
Answer:
-7 i + 16
Step-by-step explanation:
Simplify the following:
(-2 i + 1) (5 i + 6)
(1 - 2 i) (6 + 5 i) = (1) (6) + (1) (5 i) + (-2 i) (6) + (-2 i) (5 i):
6 + 5 i - 2 i×6 - 2 i×5 i
-2×6 = -12:
6 + 5 i + -12 i - 2 i×5 i
-2 i×5 i = -2 i^2 5:
6 + 5 i - 12 i + -2 i^2×5
i^2 = -1:
6 + 5 i - 12 i - 2-1×5
-2 (-5) = 10:
6 + 5 i - 12 i + 10
6 + 5 i - 12 i + 10 = (6 + 10) + (5 i - 12 i) = 16 - 7 i:
Answer: -7 i + 16
Answer: your answer to this question would be 16-7i
Step-by-step explanation:
Полоску бумаги разрезали на 7 частей. После этого самую большую
из полученных частей снова разрезали на 7 частей. Затем снова самую большую
из полученных частей разрезали на 7 частей. Так поступили много раз: на
каждом шаге самую большую часть разрезали на 7 частей. Могло ли в итоге
получиться 500 частей?
Answer:
2556285=68358556
Step-by-step explanation:
+58385-5838688(68589)8999=9858288
5 1/2 x 1 1/4
put in simplest form
Answer:
6 7/8
Step-by-step explanation:
First we would need to turn the 5 1/2 into a improper fraction. Then, do the same for the 1 1/4 as well. To do this, multiply the denominator by the whole number and then add the numerator. Once you get those values, you simply multiply across, meaning the numerators get multiplied together and the denominators get multiplied together. The value you get is the improper fraction form of the simplest form. Then you can convert it back to a mixed fraction if that is the form you need. To do this divide the numerator by the denominator. Now we have a value of 11/2 x 5/4 = 55/8. So divide 55 by 8 and leave the remainder in fraction form. Therefore the answer should be 6 7/8, after it is reduced.
I hope this helps, have a blessed day! :)
Solve the equation by the zero factor property
b^2-8b+15=0
Answer:
(b-5)(b-3)
Step-by-step explanation:
To factor this equation we have to find 2 numbers that combine to equal -8 and multiply together to equal 15
these numbers would be -3 and -5
after that you just put them in an equation like so:
(b-5)(b-3)
you can check your work by just multiplying those back together
If you want future explanation on how to do this just ask
hope this helps :D
Kristen goes on a cave tour with her family. They climb down to 8 meters below ground level.
Then they climb the opposite of -8 meters to return to ground level.
How many meters did they climb in all? Enter your answer in the box
Answer:
16 meters is the answer of this question
The radius of a circle is 12 inches. What is
the diameter?
Answer:
diameter is 24 inches
Step-by-step explanation:
diameter is basically two times the radius
Kenneth measured a hotel and made a scale drawing. The scale he used was 1 inch = 4 feet. The actual length of a room in the hotel is 20 feet. How long is the room in the drawing
Answer:
The room is 5 inches in the drawing.
Step-by-step explanation:
If the room is 20 feet and each inch in his drawing measures 4 feet what you need to do is see how many times 4 goes into 20.
4 times 5 = 20
So the answer would be 5
Hope it helps! =D
Rewrite using an exponent.
Answer:
2[tex] {}^{5} [/tex]
Step-by-step explanation:
2×2×2×2×2 = 2[tex] {}^{5} [/tex]
A plumber has a 20Foot
Answer:
14
Step-by-step explanation:
I think the question is whether: "a plumber has a 20-foot piece of PVC pipe. how many 7/5 foot pieces can be cut from the 20-foot piece?"
help please rn please help
Answer: the answer is MO=5
Step-by-step explanation:
At Keller's Bike Rentals, it costs $15 to rent a bike for 4 hours.
How many dollars does it cost per hour of bike use?
Answer:
3.75
Step-by-step explanation:
15/4=3.75
Answer:
12 1/3
Step-by-step explanation:
because when you find the total cost plus divide it gives u the answer
hans bought three books at a bookstore. Here are the prices(in dollars). 6,19.1,6.97 what is the total amount Hans paid at the bookstore
Answer:
$32.07
Step-by-step explanation:
Just add them together.
The sum of three consecutive odd numbers is 177. Find the number. Step by step please
Answer:
57, 59 and 61
Step-by-step explanation:
Let the first odd number be n The next consecutive odd number is n + 2 and the next one after that is n + 4
(consecutive odd numbers have difference of 2 between them. For example if 7 is the first odd number next one is 9(7+2) and the next one is 11(7+4)
We are given that the sum of the numbers is 177. So we have the following equation:
n + (n + 2) + (n +4) = 177
Simplifying gives
n + n + 2 + n + 4 = 177
Collect like terms:
n + n + n + 2 + 4 = 177
3n + 6 = 177
Subtract 6 from both sides
==> 3n = 177 - 6
==> 3n = 171
Divide by 3 both sides
3n/3 = 171/3
n = 57 and this is the first of the odd numbers
The next odd number is 57+ 2 = 59
and the next one after that is 59 + 2 = 61 (same as 57 + 4)