So, my buddy is new to doing Calculus and needs help understanding this equation, it would be very appeciated for some help
Answers
To evaluate the integral, rewrite the integrand as
[tex]x^{-x}=e^{\ln x^{-x}}=e^{-x\ln x}[/tex]
Recall that
[tex]e^x=\displaystyle\sum_{n=0}^\infty\frac{x^n}{n!}\implies x^{-x}=\sum_{n=0}^\infty\frac{(-x\ln x)^n}{n!}[/tex]
The leftmost sum is the well-known power series expansion for the function [tex]f(x)=e^x[/tex]. In the rightmost sum, we just replace [tex]x[/tex] with [tex]-x\ln x[/tex].
This particular power series has a property called "uniform convergence". Roughly speaking, it's a property that says a sequence of functions [tex]f_n(x)[/tex] converges to some limiting function [tex]f(x)[/tex] in the sense that [tex]f_n(x)[/tex] and [tex]f_{n+1}(x)[/tex] get arbitrarily close to one another. If you have an idea of what "convergence" alone means, then you can think of "uniform convergence" as a more powerful form of convergence.
Long story short, this property allows us to interchange the order of summation/integration to write
[tex]\displaystyle\int_0^1x^{-x}\,\mathrm dx=\int_0^1\sum_{n=0}^\infty\frac{(-x\ln x)^n}{n!}\,\mathrm dx=\sum_{n=0}^\infty\frac{(-1)^n}{n!}\int_0^1(x\ln x)^n\,\mathrm dx[/tex]
The integral can be tackled with a substitution,
[tex]x=e^{-u/(n+1)}\implies-(n+1)\ln x=u\implies\mathrm dx=-\dfrac1{n+1}e^{-u/(n+1)}\,\mathrm du[/tex]
so that the integral is equivalent to
[tex]\displaystyle\int_0^1(x\ln x)^n\,\mathrm dx=\int_\infty^0\left(e^{-u/(n+1)}\right)^n\left(-\frac u{n+1}\right)^n\left(-\frac1{n+1}e^{-u/(n+1)}\right)\,\mathrm du[/tex]
[tex]=\displaystyle\frac{(-1)^n}{(n+1)^{n+1}}\int_0^\infty e^{-u}u^n\,\mathrm du[/tex]
The remaining integral reduces to [tex]n![/tex], which you can derive for yourself via integration by parts/power reduction.
So we have
[tex]\displaystyle\int_0^1x^{-x}\,\mathrm dx=\sum_{n=0}^\infty\frac{(-1)^n}{n!}\cdot\frac{(-1)^nn!}{(n+1)^{n+1}}=\sum_{n=0}^\infty\frac1{(n+1)^{n+1}}[/tex]
which is the same as
[tex]\displaystyle\sum_{n=1}^\infty\frac1{n^n}=\sum_{n=1}^\infty n^{-n}[/tex]
and hence the identity.
Related Questions
Calculate the exact value of ( 4 1/3 - 1 2/5 ) ÷ 4/15
Answers
the answer is 11.
use BEDMAS, and calculate the equation inside the brackets (answer to that is 44/15)
then divide it by 4/15 and you get 11
let's firstly convert the mixed fractions to improper fractions and proceed.
[tex]\bf \stackrel{mixed}{4\frac{1}{3}}\implies \cfrac{4\cdot 3+1}{3}\implies \stackrel{improper}{\cfrac{13}{3}}~\hfill \stackrel{mixed}{1\frac{2}{5}}\implies \cfrac{1\cdot 5+2}{5}\implies \stackrel{improper}{\cfrac{7}{5}} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\bf \stackrel{\textit{recall that by PEMDAS, parenthesis first}~\hfill }{\left(\cfrac{13}{3}-\cfrac{7}{5} \right)\div \cfrac{4}{15}\implies \left(\stackrel{\textit{using an LCD of 15}}{\cfrac{(5)13-(3)7}{15}} \right)\div \cfrac{4}{15}\implies \left(\cfrac{65-21}{15} \right)\div \cfrac{4}{15}} \\\\\\ \left(\cfrac{44}{15}\right)\div \cfrac{4}{15}\implies \cfrac{44}{15}\times \cfrac{15}{4}\implies \cfrac{44}{4}\cdot \cfrac{15}{15}\implies 11\cdot 1\implies 11[/tex]
what the hell is a rotation?
Answers
Answer:
A rotation is when a shape rotates across the x or y axis on a coordinate plane :)
Step-by-step explanation:
The average battery life of 2800 manufactured cell phones is recorded And normally distributed. The mean battery life is 14 hours with a standard deviation of .5 hours. Find the number of phones who have a battery life in the 13 to 14 range.
Answers
Answer:
1336 phones
Step-by-step explanation:
The average battery life of 2800 manufactured cell phones has a normal distribution.
The mean battery life is 14 hours, therefore: μ = 14 hours.
The standard deviation is 0.5 hours, therefore: σ = 0.5 hours.
To find the number of phones who have a battery life in the 13 to 14 range, we're going to ask for the help of a calculator. The probability of finding phones who have a battery life in the 13 to 14 range is: 0.4772. (See attached picture)
Therefore, the number of phones who have a battery life in the 13 to 14 range is: 0.4772×2800 = 1336,16 ≈ 1336 phones.
Final answer:
To find the number of phones with a battery life between 13 and 14 hours, calculate the z-scores, find the corresponding percentage using the standard normal distribution, and then multiply by the total number of phones, resulting in approximately 1336 phones.
Explanation:
The question asks for the number of cell phones with a battery life in the 13 to 14 hour range out of 2800 phones where the mean battery life is 14 hours with a standard deviation of 0.5 hours. The battery life is normally distributed.
To find this number, we need to calculate the z-scores for 13 and 14 hours and then determine the percentage of phones between those z-scores. The z-score for 13 hours is (13 - 14)/0.5 = -2 and for 14 hours is (14 - 14)/0.5 = 0. Using the standard normal distribution table, the area under the curve from -2 to 0 is approximately 47.7%. Therefore, the number of phones with battery life between 13 and 14 hours is 0.477 * 2800 ≈ 1336 phones.
which matrix is equal to [-6,-6.5,1.7,2,-8.5,19.3]
Answers
Answer:
Bruh its D.
Step-by-step explanation:
D has the exact same values as the original matrix.
Answer:
[tex]\large\boxed{\left[\begin{array}{ccc}-6&-6.5&1.7\\2&-8.5&19.3\end{array}\right]}[/tex]
Step-by-step explanation:
Equal matrices are identical. We have the same numbers in the same places. Therefore, the matrix equal to a given matrix is the same matrix.
Olivia solved a division problem by subtracting and keeping track as shown. Identify the problem she solved.
Answers
Answer: 314÷13
..............................................................................
Answer:
13/314
Step-by-step explanation:
The answer and how to do it for me homework
Answers
Answer:
15
Step-by-step explanation:
Answer:
y = -5
Step-by-step explanation:
You are solving for y. You want to know which value of y makes the equation true. You must end up with y alone on the left side equaling a number.
12y + 25 = -35
First, subtract 25 from both sides.
12y + 25 - 25 = -35 - 25
12y = -60
Now divide both sides by 12.
12y/12 = -60/12
y = -5
Answer: y = -5
The graph below plots the values of y for different values of x:
Plot the ordered pairs 1, 8 and 2, 3 and 3, 0 and 4, 1 and 5, 2 and 6, 1
What is the correlation coefficient for the data plotted?
−0.9
−0.6
0.6
0.9
Answers
Answer:
-0.6
Step-by-step explanation:
we know that
The correlation coefficient is a measure of the strength of the straight-line or linear relationship between two variables. Correlation coefficients are expressed as values between +1 and -1
we have
(1,8),(2,3),(3,0),(4,1),(5,2),(6,1)
using a Excel tool (CORREL function)
see the attached table
The correlation coefficient is -0.68
Answer:
X Values
∑ = 21
Mean = 3.5
∑(X - Mx)2 = SSx = 17.5
Y Values
∑ = 15
Mean = 2.5
∑(Y - My)2 = SSy = 41.5
X and Y Combined
N = 6
∑(X - Mx)(Y - My) = -18.5
R Calculation
r = ∑((X - My)(Y - Mx)) / √((SSx)(SSy))
r = -18.5 / √((17.5)(41.5)) = -0.6865
Meta Numerics (cross-check)
r = -0.6865
what is the square root of m to the power of 6
Answers
Step-by-step explanation:
[tex]\sqrt{m^6}=\sqrt{m^{3\cdot2}}\qquad\text{use}\ (a^n)^m=a^{nm}\\\\=\sqrt{(m^3)^2}\qquad\text{use}\ \sqrt{a^2}=|a|\\\\=|m^3|\\\\\text{if}\ m\geq0,\ \text{then}\ \sqrt{m^6}=m^3\\\\\text{if}\ m<0,\ \text{then}\ \sqrt{m^6}=-m^3[/tex]
In exercises 18 and 19 determine which solution if any is an extraneous solution 18.sprt(3x-2)=x; x=1,x=2. 19. Sprt(x+6=x; x=3,x=-2
Answers
ANSWER
18. No extraneous solution.
19. The extraneous solution is x=-2
EXPLANATION
18. The given radical equation is:
[tex] \sqrt{3x - 2} = x[/tex]
Solving this radical equation yields
[tex]x=1,x=2[/tex]
We check for an extraneous solution by substituting each value into the equation.
Checking for x=1,
[tex] \sqrt{3 \times 1 - 2} = 1[/tex]
[tex]\sqrt{3- 2} = 1[/tex]
[tex]\sqrt{1} = 1[/tex]
[tex]1 = 1[/tex]
This is true.
Checking for x=2
[tex]\sqrt{3 \times 2- 2} = 2[/tex]
[tex]\sqrt{6- 2} = 2[/tex]
[tex]\sqrt{4} = 2[/tex]
[tex]2 = 2[/tex]
This is also true. Hence there is no extraneous solution.
19. The given radical equation is:
[tex] \sqrt{x + 6} = x[/tex]
Solving this equation yields,
[tex]x=3,x=-2[/tex]
Checking for x=3,.
[tex]\sqrt{3+ 6} = 3[/tex]
[tex] \sqrt{9} = 3[/tex]
3=3.
This is a true solution.
Checking for x=-2.
[tex]\sqrt{ - 2 + 6} = - 2[/tex]
[tex] \sqrt{4} = - 2[/tex]
[tex]2 \ne - 2[/tex]
Hence x=-2 is an extraneous solution.
please help and look at the picture
Answers
ANSWER
A.
[tex] \frac{1}{64} [/tex]
EXPLANATION
The given expression is:
[tex] {4}^{ - \frac{11}{3} } \div {4}^{ - \frac{2}{3} } [/tex]
Recall that:
[tex] {a}^{m} \div {a}^{n} = {a}^{m - n} [/tex]
We apply this property to obtain:
[tex]{4}^{ - \frac{11}{3} } \div {4}^{ - \frac{2}{3} } = {4}^{ - \frac{11}{3} - - \frac{2}{3} } [/tex]
Collect LCM
[tex]{4}^{ - \frac{11}{3} } \div {4}^{ - \frac{2}{3} } = {4}^{ \frac{ - 11 + 3}{3}} [/tex]
Simplify;
[tex]{4}^{ - \frac{11}{3} } \div {4}^{ - \frac{2}{3} } = {4}^{ \frac{ - 9}{3}} [/tex]
.
[tex]{4}^{ - \frac{11}{3} } \div {4}^{ - \frac{2}{3} } = {4}^{ - 3} [/tex]
[tex]{4}^{ - \frac{11}{3} } \div {4}^{ - \frac{2}{3} } = \frac{1}{ {4}^{3} } [/tex]
[tex]{4}^{ - \frac{11}{3} } \div {4}^{ - \frac{2}{3} } = \frac{1}{64} [/tex]
The first choice is correct
The vertex of this parabola is at (-1,3) Which of the following could be its equation
Answers
Answer:
y - 3 = a(x + 1)^2
Step-by-step explanation:
The vertex equation of a vertical parabola is:
y - k = a(x - h)^2.
If the vertex is at (-1, 3), then the equation becomes:
y - 3 = a(x + 1)^2, where a is a constant.
Next time, would you please share the answer choices. Thank you.
Answer:
Step-by-step explanation:
all parabola have equation : y = a(x +1)²+3 a in R
8x-6y=54 in slope intercept form please!!
Answers
Answer: y= 4/3x-9
Step-by-step explanation:
The equation 8x - 6y = 54 can be converted to slope-intercept form (y = mx + b) by isolating y. The steps involve rearranging terms and simplifying to yield the equation: y = (4/3)x - 9.
Explanation:First, you want to manipulate your equation: 8x - 6y = 54 into slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. Here are the steps:
Add 6y to both sides to isolate variables on one side. Now, the equation is 8x = 6y + 54. Divide everything by 6 to solve for y. The equation becomes y = (8x/6) - (54/6). Simplify the equation: y = (4/3)x - 9.
Therefore, the equation 8x - 6y = 54 in slope intercept form is y = (4/3)x - 9.
Learn more about Slope-Intercept Form here:
https://brainly.com/question/29146348
#SPJ3
How do I do this algebra??
Answers
Answer:
75
Step-by-step explanation:
Square both sides. 225=3x^2
divide both sides by 3
x^2=75
x=√75= 5√3
[tex]15 = x \sqrt{3} \\ \\ 1. \: 15 = \sqrt{3x} \\ 2. \: \frac{15}{ \sqrt{3} } = x \\ 3. \: x = \frac{15}{ \sqrt{3} } [/tex]
math help ^^ will give 10 points ^-^
Answers
Answer:
C
Step-by-step explanation:
Arc length is:
s = 2πr (θ/360)
Given θ = 80° and r = 7 ft:
s = 2π (7 ft) (80 / 360)
s = 1120π/360
s = 28π/9
Answer is C.
Which of these expressions is equivalent to log (20/3)
Answers
Answer:
C log (20) - log (3)
Step-by-step explanation:
log (a/b) = log (a) - log (b)
log (20/3) = log (20) - log (3)
Answer: C. log (20) - log (3)
Step-by-step explanation: a p e x /\ dude above me is right too
I only need someone to put it in the quadratic formula I already have solutions. Please answer ASAP!!! I will give brainliest!!!
Answers
The quadratic formula is [tex]\frac{-b+-\sqrt[2]{b^2-4ac}}{2a}[/tex]
and in the equation ax^2+bx+c=0
so now all you have to do is substitute the numbers into the quadratic formula
Which of the following is the order of magnitude for the number feet in a mile A.4
B.3
C.1
D.2
Answers
Answer:
b
Step-by-step explanation:
What is the meaning of Zero on a number line
Answers
The definition of a number line is a straight line with a "zero" point in the middle, with positive and negative numbers listed on either side of zero and going on indefinitely.
What percent of a dozen is 3
Answers
Answer:
25%
Step-by-step explanation:
Percentages are one of several ways of describing quantities' relationships to one another. Specifying one number as a percentage of another means specifying the fraction of the second quantity the first comprises. The percentage value is the number that, divided by 100, equals that fraction. To express the percentage as a whole number, round it accordingly. Some applications, however, don't require percentages as exact whole figures.
Divide the first number the second. For instance, if you want to find what percentage 43 is out of 57, divide 43 by 57 to get 0.754386.
Which geometric series converges ???
Answers
Answer:
C
Step-by-step explanation:
A geometric series will only converge if - 1 < r < 1
sum to infinity = [tex]\frac{a}{1-r}[/tex]
The nth term formula for a geometric series is
[tex]a_{n}[/tex] = a[tex]r^{n-1}[/tex]
where a is the first term and r the common ratio
The only summation with - 1 < r < 1 is C where r = - 0.2
Answer: The correct option is
(C) [tex]\sum_{n=1}^{\infty}4(-0.2)^{n-1}.[/tex]
Step-by-step explanation: We are give to select the geometric series that converges.
We know that
the general (n-th) term of a common geometric series is given by
[tex]a_n=ar^{n-1}.[/tex]
And the series converges if the modulus of the common ratio is less than 1, .e., |r| < 1.
Now, for the first infinite geometric series, we have
[tex]a_n=\dfrac{2}{3}(-3)^{n-1}.[/tex]
So, the common ratio will be
[tex]r=-3~~~\Rightarrow |r|=3>1.[/tex]
That is, the series will not converge. Option (A) is incorrect.
For the second geometric series, we have
[tex]a_n=5(-1)^{n-1}.[/tex]
So, the common ratio will be
[tex]r=-1~~~\Rightarrow |r|=1.[/tex]
That is, the series will not converge. Option (B) is incorrect.
For the third geometric series, we have
[tex]a_n=4(-0.2)^{n-1}.[/tex]
So, the common ratio will be
[tex]r=-0.2~~~\Rightarrow |r|=0.2<1.[/tex]
That is, the series will CONVERGE. Option (C) is correct.
For the fourth geometric series, we have
[tex]a_n=0.6(-2)^{n-1}.[/tex]
So, the common ratio will be
[tex]r=-2~~~\Rightarrow |r|=2>1.[/tex]
That is, the series will not converge. Option (D) is incorrect.
Thus, (C) is the correct option.
Determine if the binomial is a perfect square binomial. If so, show the original monomial squared.
5. x2+ 16
6. x4 + 12x2+ 36
Answers
Answer:
Step-by-step explanation:
(5) x^2 + 16 is a perfect square binomial only if imaginary roots are allowed.
x^2 + 16 = (x + 4i)(x - 4i)
(6) x^4 + 12x^2 + 36 is a perfect square trinomial.
The square root of x^4 is x^2 and the square root of 36 is 6.
Experimenting, we find that x^4 + 12x^2 + 36 = (x² + 6)²
so we can conclude that x^4 + 12x^2 + 36 = (x² + 6)(x² + 6) = (x² + 6)².
Please need help
Find area of shaded region.
Round to the nearest tenth
Answers
Answer:
818.4 in²
Step-by-step explanation:
The area (A) of the shaded region is
A = area of circle - ( area of white sector + area of triangle )
= ( π × 27.8²) - (π × 27.8² × [tex]\frac{210}{360}[/tex] +(0.5 × 27.8 ×27.8 × sin150°)
= 2427.95 - (1416.30 + 193.21 )
= 2427.95 - 1609.51 ≈ 818.4 in²
i am a number. If the area of each square below is 25 square units, i am the perimeter of the figure. i am ___________
Answers
Answer: 20
Step-by-step explanation:
The area= 25
The square root of 25=5
So we know that 2 sides are 5
5×4=20
Multiply 5 by 4 because squares have all sides of the same length, so the remaining 2 sides are 5.
Please help and thank you
Answers
Answer:
C. or D. I would say D though.
Step-by-step explanation:
All you have to do is look at the total with both children and adults, then you can take that and divide it by the total of either children or adults and then you would get the answer D.
Answer: B
Step-by-step explanation:
One side of a rectangle is 3 feet shorter than twice the other side find the sides if the area is 209 feet squared
Answers
Answer:
29119.66666...+14561.3333...=43681, 209ft^2, not square feet.
Step-by-step explanation:
"One side of a rectangle is 3 feet shorter than twice the other side find the sides if the area is 209 feet squared"
Ok, so this is a tricky one- 209 feet squared, not 209 square feet, therefore the area is 209^2, or 43,681.
Next, let's define our variables-
x= the "other side"
z= 3 feet shorter than twice side
We can now make these (useful) equations
z=2x-3
43681= (2x-3)+x
We will focus on the latter for now-
Simplify
43681= (2x-3)+x
43681= 2x-3+x
43681= 3x-3
+3
43684=3x
/3
14561.3333...=x
z=2x-3
z=2*(14561.3333)-3
z=29119.66666....
29119.66666...+14561.3333...=43681
the radius of a regular pentagon measures 7ft. which of the following would be an acceptable equation to use to solve for the measure of the side, s?
(the selected answer is just randomly selected)
Answers
Answer:
Option D. [tex]cos(54\°)=\frac{0.5s}{7}[/tex]
Step-by-step explanation:
we know that
A regular pentagon can be divided into 5 isosceles triangles
The length side of the legs of one isosceles triangle is equal to the radius
The vertex angle of one isosceles triangle is equal to 360/5=72 degrees
The base angle of one isosceles triangle is equal to 54 degrees
Let
s------> the length side of the regular pentagon
so
[tex]cos(54\°)=\frac{(s/2)}{r}[/tex]
substitute the given value
[tex]cos(54\°)=\frac{(s/2)}{7}[/tex]
[tex]cos(54\°)=\frac{0.5s}{7}[/tex]
Give the polynomial function below find f(-4) f(x) =2x^2 - x+ 9
Answers
Answer:
f(-4) = 45Step-by-step explanation:
Put x = -4 to f(x) = 2x² - x + 9:
f(-4) = 2(-4)² - (-4) + 9 = 2(16) + 4 + 9 = 32 + 4 + 9 = 45
whats the y intercept to x g(x) 0 2 1 6 2 10
Answers
Answer:
y-intercept = (0,2).
Step-by-step explanation:
We have been given a table of the function g(x) as shown below:
x g(x)
0 2
1 6
2 10
Using that table we need to find about what is the y-intercept of the given function g(x).
By definition of y-intercept, we know that, y-intercept is the y-value or function value of g(x) value when x=0.
From table we see that g(x)=2 when x=0.
So the y-intercept = 2.
In point form we can write y-intercept as (0,2).
please help with this thank you
Answers
Answer:
(4, - 2 )
Step-by-step explanation:
To find the x- coordinate substitute y = - 2 into the equation
y + 2 = - 3(x - 4), thus
- 2 + 2 = - 3x + 12
0 = - 3x + 12 ( subtract 12 from both sides )
- 12 = - 3x ( divide both sides by - 3)
4 = x
Write the equation of a line in slope intercept form that is parallel to 2X plus 4Y equals 10 and passes through the point (8,2)
Answers
Answer: [tex]y=-\frac{1}{2}x+6[/tex]
Step-by-step explanation:
The equation of the line is slope-intercept form is:
[tex]y=mx+b[/tex]
Where m is the slope and b thte y-intercept.
The lines are parallel, then they have the same slope.
Solve for "y" from [tex]2x+4y=10[/tex] to find the slopes of the lines :
[tex]2x+4y=10\\4y=-2x+10\\y=-\frac{1}{2}x+\frac{5}{2}[/tex]
The value of the slopes of the lines is:
[tex]m=-\frac{1}{2}[/tex]
Substitute the slope and the point into the equation of the line and solve for "b":
[tex]2=-\frac{1}{2}(8)+b\\2=-4+b\\b=6[/tex]
Then the equation of this line is:
[tex]y=-\frac{1}{2}x+6[/tex]
The answer is:
The equation of the new line will be:
[tex]y=-0.5x+6[/tex]
or
[tex]y=-\frac{1}{2}x+6[/tex]
Why?To solve the problem, we need to remember the slope intercept form of a line.
The slope intercept form of a line is given by the following equation:
[tex]y=mx+b[/tex]
Where,
y, is the function.
x, is the variable of the function.
m, is the pendant of the line.
b, is the y-axis intercept of the line.
So, we are given the line that will be parallel to the line that we are looking for:
[tex]2x+4y=10\\4y=-2x+10\\4y=-2(x-5)\\y=\frac{-2}{4}*(x-5)\\\\y=-\frac{1}{2}*(x-5)\\\\y=-\frac{1}{2}x+\frac{5}{2}[/tex]
Where,
[tex]m=-\frac{1}{2}[/tex]
Then,
We need to use the same slope to guarantee that the new line will be parallalel to the given line-
So, our new line will have the following form:
[tex]y=-\frac{1}{2}x+b[/tex]
We need to substitute the given point to isolate "b" in order to guarantee that the line will pass through.
Now, substituting the given point, to calculate"b", we have:
Calculating b, we have:
[tex]2=-\frac{1}{2}8+b[/tex]
[tex]2=-4+b[/tex]
[tex]2+4=b[/tex]
[tex]6=b[/tex]
Hence, we have that the equation of the new line will be:
[tex]y=-0.5x+6[/tex]
or
[tex]y=-\frac{1}{2}x+6[/tex]
Proving that the line will pass through the given point, by substituting it into its equation, we have:
[tex]2=-0.5(8)+6[/tex]
[tex]2=-4+6[/tex]
[tex]2=2[/tex]
So, since the equality is satisfied, we know that the line pass through the new line.
Have a nice day!