Let [tex]S[/tex] denote the sum. We can first resolve the sum in [tex]m[/tex] by factorizing and decomposing into partial fractions.
[tex]\displaystyle S = \sum_{n=1}^\infty \sum_{m=1}^\infty \frac{(-1)^{n+1}}{mn^2 + mn + m^2n} \\\\ ~~~~ = \sum_{n=1}^\infty \frac{(-1)^{n+1}}n \sum_{m=1}^\infty \frac1{m(m+n+1)} \\\\ ~~~~ = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n(n+1)} \sum_{m=1}^\infty \left(\frac1m - \frac1{m+n+1}\right)[/tex]
Rewrite the [tex]m[/tex]-summand as a definite integral. Interchange the integral and sum, and evaluate the resulting geometric sums.
[tex]\displaystyle \sum_{m=1}^\infty \left(\frac1m - \frac1{m+n+1}\right) = \sum_{m=1}^\infty \int_0^1 \left(x^{m-1} - x^{m+n}\right) \, dx \\\\ ~~~~~~~~ = \int_0^1 \sum_{m=1}^\infty \left(x^{m-1} - x^{m+n}\right) \, dx \\\\ ~~~~~~~~ = \int_0^1 \frac{1 - x^{n+1}}{1 - x} \, dx \\\\ ~~~~~~~~ = \int_0^1 \sum_{\ell=0}^n x^\ell \, dx \\\\ ~~~~~~~~ = \sum_{\ell=0}^n \int_0^1 x^\ell \, dx \\\\ ~~~~~~~~ = \sum_{\ell=0}^n \frac1{\ell+1} \\\\ ~~~~~~~~ = \sum_{\ell=1}^{n+1} \frac1\ell = H_{n+1}[/tex]
where
[tex]H_n = \displaystyle \sum_{\ell=1}^n \frac1\ell = 1 + \frac12 + \frac 13 + \cdots + \frac1n[/tex]
is the [tex]n[/tex]-th harmonic number. The generating function will be useful:
[tex]\displaystyle \sum_{n=1}^\infty H_n x^n = -\frac{\ln(1-x)}{1-x}[/tex]
To evaluate the remaining sum to get [tex]S[/tex], let
[tex]\displaystyle f(x) = \sum_{n=1}^\infty \frac{H_{n+1}}{n(n+1)} x^{n+1}[/tex]
and observe that [tex]S=\lim\limilts_{x\to-1^+} f(x)[/tex], which I'll abbreviate to [tex]f(-1)[/tex]. Differentiating twice, we have
[tex]\displaystyle f'(x) = \sum_{n=1}^\infty \frac{H_{n+1}}n x^n[/tex]
[tex]\displaystyle f''(x) = \sum_{n=1}^\infty H_{n+1} x^n[/tex]
[tex]\displaystyle \implies f''(x) = -\frac{\ln(1-x)}{x^2(1-x)} - \frac1x[/tex]
By the fundamental theorem of calculus, noting that [tex]f(0)=f'(0)=0[/tex], we have
[tex]\displaystyle \int_{-1}^0 f'(x) \, dx = f(0) - f(-1) \implies f(-1) = -\int_{-1}^0 f'(x) \, dx[/tex]
[tex]\displaystyle \int_x^0 f''(x) \, dx = f'(0) - f'(x) \implies f'(x) = -\int_x^0 f''(t) \, dt[/tex]
[tex]\displaystyle \implies S = f(-1) = \int_{-1}^0 \int_x^0 \left(\frac{\ln(1-t)}{t^2(1-t)} + \frac1t\right) \, dt \, dx[/tex]
Change the order of the integration, and substitute [tex]t=-u[/tex].
[tex]S = \displaystyle \int_{-1}^0 \int_{-1}^t \left(\frac{\ln(1-t)}{t^2(1-t)} + \frac1t\right) \, dx \, dt \\\\ ~~~ = - \int_{-1}^0 \left(\frac{(1+t) \ln(1-t)}{t^2(1-t)} + \frac1t + 1\right) \, dt \\\\ ~~~ = -1 - \int_{-1}^0 \left(\left(\frac2{1-t} + \frac2t + \frac1{t^2}\right) \ln(1-t) + \frac1t\right) \, dt \\\\ ~~~ = -1 - \int_0^1 \left(\left(\frac2{1+u} - \frac2u + \frac1{u^2}\right) \ln(1+u) - \frac1u\right) \, du[/tex]
For the remaining integrals, substitute and use power series.
[tex]\displaystyle \int_0^1 \frac{\ln(1+u)}{1+u} \, du = \int_0^1 \ln(1+u) d(\ln(1+u)) = \frac{\ln^2(2)}2[/tex]
[tex]\displaystyle \int_0^1 \frac{\ln(1+u)}u \, du = - \int_0^1 \frac1u \sum_{k=1}^\infty \frac{(-u)^k}k \, du \\\\ ~~~~~~~~~~~~~~~~~~~~~~ = - \sum_{k=1}^\infty \frac{(-1)^k}k \int_0^1 u^{k-1} \, du \\\\ ~~~~~~~~~~~~~~~~~~~~~~ = - \sum_{k=1}^\infty \frac{(-1)^k}{k^2} = \frac{\pi^2}{12}[/tex]
[tex]\displaystyle \int_0^1 \frac{\ln(1+u) - u}{u^2} \, du = - \int_0^1 \frac1{u^2} \left(\sum_{k=1}^\infty \frac{(-u)^k}k + u\right) \, du \\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = -\int_0^1 \frac1{u^2} \sum_{k=2}^\infty \frac{(-u)^k}k \, du \\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = - \sum_{k=2}^\infty \frac{(-1)^k}k \int_0^1 u^{k-2} \, du \\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = -\sum_{k=2}^\infty \frac{(-1)^k}{k(k-1)} \\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \sum_{k=1}^\infty \frac{(-1)^k}{k(k+1)} = 1 - 2\ln(2)[/tex]
Tying everything together, we end up with
[tex]S = -1 - \left(2 \cdot \dfrac{\ln^2(2)}2 - 2 \cdot \dfrac{\pi^2}{12} + (1-2\ln(2))\right) \\\\ ~~~ = \boxed{\frac{\pi^2}6 - 2 + 2\ln(2) - \ln^2(2)}[/tex]
COULD YALL HELP ME ?!???!
Answer:
28 degrees
Step-by-step explanation:
We know that (4x + 7) + (2x + 5) add up to a straight angle = 180 degrees, so we have the equation 4x + 7 + 2x + 5 = 180.
By combining like terms, we get 6x + 12 = 180.
Subtract 12 from both sides of the equation to get 6x = 168. Divide both sides by 6 and you get x = 28.
What is the volume of a cube with
1/3 inch sides?
Answer:
1/27 in^3.
Step-by-step explanation:
1/3 * 1/3 * 1/3
Find the value of a and b, so that the system of equations has a solution (2, -3)
3x + 5y + 9 = 0
ax + by = 11
Please give an explanation with the answer, thats what im most worried about. 100 points :3
please help right answer will get brainliest
Answer:
[tex]\sf y = \dfrac{x }{\boxed{26}} +\boxed{-2}[/tex]
Step-by-step solution:
[tex]\rightarrow \sf x - 26y = 52[/tex]
[tex]\rightarrow \sf - 26y = 52-x[/tex]
[tex]\rightarrow \sf - 26y = -x + 52[/tex]
[tex]\rightarrow \sf y = \dfrac{-x + 52}{-26}[/tex]
[tex]\rightarrow \sf y = \dfrac{-x }{-26} + \dfrac{ 52}{-26}[/tex]
[tex]\rightarrow \sf y = \dfrac{x }{26} -2[/tex]
A new Community Sports Complex is being built in Del Rio. The perimeter of the rectangular playing field is 354 yards.
The length of the field is 7 yards less than triple the width. What are the dimensions of the playing field?
We need to know about the perimeter of a rectangle to solve this problem.The length of the field is 131 yards and width is 46 yards.
The perimeter of a rectangle is the summation of it's length and breadth, the longer side of a rectangle is called length and the shorter side is called breadth. The perimeter of a rectangle is given by the formula 2(l+b) where l is the length of the rectangle and b is the breadth of the rectangle. In this question we have been told the perimeter of the rectangular field is 354 yards. The length of the field is 7 yards less than triple the width. Let the width of the field be x yard, then the length is
length=3x-7
Then perimeter in terms of x= 2(l+b)=2((3x-7)+x)=2(4x-7)=8x-14
8x-14=354
8x=368
x=46
Then breadth=46 yards
length =3x-7=138-7=131 yards
Therefore we found out the dimensions of the field to be, the length is 131 yards and breadth is 46 yards.
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distance between (4,9) (9,2)
The distance between the two points is 8.6 units
How to determine the distance between the two points?The points are given as
(4, 9) and (9, 2)
The distance between the two points is then calculated as
d = √(x2 - x1)^2 + (y2 - y1)^2
This gives
d = √(4 - 9)^2 + (9 - 2)^2
Evaluate the sum of exponents
So, we have
d = √74
Evaluate the exponents
So, we have
d = 8.6
Hence, the distance between the two points is 8.6 units
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Kai is buying boxes of cookies at a bakery. Each box of cookies costs \$4$4dollar sign, 4. He wrote this equation to find how many dollars he'll spend (d)(d)left parenthesis, d, right parenthesis when he buys some number of boxes of cookies (c)(c)left parenthesis, c, right parenthesis:
d=4cd=4cd, equals, 4, c
Identify the dependent and independent variables.
The number of boxes of cookies Kai buy : independent variable
The amount of the cookies will cost in dollars : dependent variables
Based on the given conditions, formulate,
Each box of cookies costs is = \$4$4dollar (sign, 4).
Cookie represented by c
The required equation is,
d = 4cd = 4cd = 4c
The price is determined by the number of cookies purchased.
c is the independent variable
when c is changes d is changed
So, d is the dependent variable
variables of the equation.
So,
We can write,
d = 4c
where d represents the cost of cookies in dollar and c represents the number of boxes of cookies.
Then, the cost of each box of cookies is $4.
It means,
The cost of 1 box of cookies is = $4.
The cost of 2 boxes of cookies is = $(4+4) = $(2×4)
The cost of 3 boxes of cookies is = $(4+4+4) = $(3×4)
The cost of cookies = (Number of boxes × 4)
The required equation is,
d = 4c
where d represents the cost of cookies in dollar and c represents the number of boxes of cookies.
Therefore,
The boxes of cookies Kai buy : independent variable
The cookies will cost in dollars : dependent variables
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Boxes of cookies (c) is Independent.
Dollars spent (d) is Dependent.
PLEASE HELP FAST I ONLY HAVE ONE HOUR!!!!!!
You are laying brick for a custom circular patio fire pit. The radius of the fire pit is 36 inches. In square feet, what
amount of patio space is needed for the fire pit?
O 9.42 square feet
O 28.26 square feet
O 113.04 square feet
226.08 square feet
4,069.44 square feet
Answer:
The answer is 28.26 square feet.
Step-by-step explanation:
The area of a circle is [tex]\pi r^{2}[/tex], so if the radius is 36, [tex]\pi *36^{2}[/tex] is equal to about 4071.5. The formula for converting square inches to square feet is to divide by 144, and [tex]\frac{4071.5}{144}[/tex] is around 28.26.
0.695 into fraction in the simplest form
Answer:
139/200
Step-by-step explanation:
.695 can be represented as 695/1000. Divide the numerator and denominator by 5, you get 139/200.
Answer: 139/200
Step-by-step explanation:
I am guessing
Help!
solve for x 13,14, and 16 only!!!
Answer:
13:
x= 86°
14:
x= 30°
16:
x= 59°
f(x)=5x-12what is f(-4)
A company's sales in Seattle were $330,000 in 2012, while their sales in Portland were $245,000 for
the same year. Complete the following statements:
a. Seattle's sales were
% larger than Portland's.
b. Portland sales were
% smaller than Seattle's.
c. Portland sales were
% of Seattle's.
find the inverse of this grapg
Answer:
Can you please send the picture of the graph
Step-by-step explanation:
The difference of two numbers is 6. Twice the smaller is 4 more than the larger. Find the larger
number
Answer:
16
Step-by-step explanation:
Difference means subtract
Let the two numbers be x and y
x-y = 6
Let y be the smaller number
2y = x+4
Using substitution
x = 6+y
2y = 6 +y +4
2y = y+10
Subtract y from each side
2y-y = y+10-y
y = 10
Now we need to find x
x-y = 6
x-10 = 6
Add 10 fo each side
x-10+10 =6+10
x = 16
tim goes on a car journey for the first 15 minutes his average speed is 40mph
Answer:
For the first 15 minutes his average speed is 40 mph. He then stops for 25 minutes. He then completed the journey at an average speed of 45 mph. The total journey time is 1 hour.
One day total attendance at the state fair was 126,945. Round 126,945 to the nearest ten thousand, nearest ten thousand and the nearest thousand. Which of these rounded amounts is closest to the actual attendance?
Answer: To the nearest thousand.
Step-by-step explanation:
First, we will round 126,945 to the requested place values;
(it says ten thousand once, so I'm going to include hundred thousand)
hundred thousand ➜ 126,945 ➜ 100,000
ten thousand ➜ 126,945 ➜ 130,000
thousand ➜ 126,945 ➜ 127,000
Next, we will subtract these values from 126,945, and see which absolute value is the smallest.
hundred thousand ➜ 126,945 - | 100,000 | = 26,945
ten thousand ➜ 126,945 - | 130,000 | = 3,050
thousand ➜ 126,945 - | 127,000 | = 55
The rounded amount that is closest to the actual attendance is thousand.
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13. The Largo
Middle School track team needs new
uniforms. The students plan to sell plush toy tigers
(the school mascot) for $5. The students find three
companies online that sell stuffed mascots.
a.
Which company has the lowest cost per tiger?
b. If they use that company, how much profit
will the students make for each tiger sold?
a) The lowest cost per tiger is $2.74, according to Company C. (Divide the costs by how many stuffed animals there are; in this case, it's $41.10/15).
b) They would make $2.26 profit for each tiger sold.
How to know which company has the lowest cost per tiger and how much profit will the students make for each tiger sold?You must divide the pricing by the anticipated number of sales to determine how much money customers must pay for each plush toy. For Company C, the calculation would be $41.10 divided by 15. You'll discover that Company C is the least expensive of all the companies when you do that.
By deducting $2.74 from $5, you can calculate their profit.
Although they had to spend $2.74 to purchase the plush animals, they are selling it for $5. $5 - $2.74 = $2.26
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Lines AB and C'D are parallel. Find the measures of the three angles in triangle ABF.
Part 1
∠FBA = 42° (alternate interior angles)
Part 2
∠FDC = 23° (linear pair)
∠BAD = 23° (alternate interior angles)
Part 3
∠AFBA = 115° (sum of angles in a triangle)
Joshua Kelly estimates that taking some classes would result in earning $11,100 more a year for the next 20 years. Based on an annual
interest rate of 8 percent, calculate the future value of these classes. Use Exhibit 1-B. (Round time value factor to 3 decimal places
and final answer to 2 decimal places.)
Answer:
$507 958.20
Step-by-step explanation:
11 100 x table value for 20 years , 8%
11,100 x 45.762 = $ 507,958.20
GIVING BRAINIEST IF CORRECT!!!!
Express in simplest radical form.
Answer: -2√10
Step-by-step explanation:
-1√10 - 1√10 = -2√10.
Since they are like terms, you'd simply just collect the two together. Think of it as just doing -1 - 1 = -2 and adding the radical behind the two.
You'd also leave it as √10 because you're unable to break it down or simplify it further than that.
Hope I was able to help you out.
[tex]6\frac{1}{3}+4\frac{3}{4}+3\frac{7}{12}=[/tex]
Answer:
14[tex]\frac{2}{3}[/tex]
Step-by-step explanation:
Make equivalent fractions with all the same denominator. Let's us 12 as our common denominator
[tex]\frac{1}{3}[/tex] x [tex](\frac{4}{4})[/tex] = [tex]\frac{4}{12}[/tex] Another name for [tex]\frac{4}{4}[/tex] is 1. (4 divided by 4 is 1) I am just making an equivalent fraction of [tex]\frac{1}{3}[/tex]
[tex]\frac{3}{4}[/tex] x [tex](\frac{3}{3})[/tex] = [tex]\frac{9}{12}[/tex] another name for [tex]\frac{3}{3}[/tex] is 1 (3 divided by 3 is 1) I am just making an equivalent fraction of [tex]\frac{3}{4}[/tex].
Now that my denominators are all the same size I can see how many 12's I have.
6 [tex]\frac{4}{12}[/tex] + 4 [tex]\frac{9}{12}[/tex] + 3 [tex]\frac{7}{12}[/tex] I can rearrange this to add all the whole numbers together and all of the fractions together.
6 + 4 + 3 = 13
[tex]\frac{4}{12}[/tex] + [tex]\frac{9}{12}[/tex] + [tex]\frac{7}{12}[/tex] = [tex]\frac{20}{12}[/tex] I have 20 12ths. every [tex]\frac{12}{12}[/tex] makes another whole, so
[tex]\frac{20}{12}[/tex] 1 whole with [tex]\frac{8}{12}[/tex] Left over. If we put this all together, we have
13 + 1 + [tex]\frac{8}{12}[/tex]
14 [tex]\frac{8}{12}[/tex] To simplify this. I will divide [tex]\frac{8}{12}[/tex] ÷ [tex]\frac{4}{4}[/tex] = [tex]\frac{2}{3}[/tex].
the final answer is 14 [tex]\frac{2}{3}[/tex]
graph each function and give its domain
f(x) = √x - 2
Domain = ∈ [2, ∞]
Range = ∈ [0, ∞]
Graph [tex]\sqrt{x - 2 [-10, 10, -5, 5]}[/tex]
Just outline the function.
The typical root operation is:
[tex]y = a * \sqrt{(x - h)} + k[/tex]
where x - h ≥ 0
The value within the root sign of a square root function CANNOT be < 0
∴ x - h ≥ 0
∴ x ≥ 2
Therefore, Domain = ∈ [2, ∞]
Regarding range, the function we were provided doesn't have a K value.
As a result, the range spans from 0 to infinity.
Therefore, Range = ∈ [0, ∞]
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4Y-4(-2z+6y)-2z simplify
The simplified expression of 4y - 4(-2z + 6y) - 2z is -20y + 6z
What are expressions?Expressions are mathematical statements that are represented by variables, coefficients and operators
How to simplify the expression?The expression is given as
4y - 4(-2z + 6y) - 2z
Open the bracket
4y - 4(-2z + 6y) - 2z = 4y + 8z - 24y - 2z
Collect the like terms
4y - 4(-2z + 6y) - 2z = 4y - 24y + 8z - 2z
Evaluate the like terms
4y - 4(-2z + 6y) - 2z = -20y + 6z
Hence, the simplified expression of 4y - 4(-2z + 6y) - 2z is -20y + 6z
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Solve - 3g-2| +1 < 6.
Answer: 18=18
Step-by-step explanation:
If Alpha and Beta are the roots of 2x² - 5x - 6=0, find the equation whose toots are ( Alpha -1) and (Beta+1).
pls help with this and lots of explanations.Tnx
Answer:
[tex]x = 28[/tex]
[tex]2x { } ^{2} - 5x - 6 = 0[/tex]
Question 6
WORLD RECORD The world's largest ice cream cake was created on May 10, 2011, in Toronto, Canada. The cake
was 4.45 meters long, 4.06 meters wide, and 1 meter tall. All surfaces of the cake except the bottom were covered
with a cookie crumble topping. Use an appropriate two-dimensional model to approximate the area covered by the
cookie crumble topping. Round your answer to the nearest tenth of a square meter.
m²
The surface area covered by the rectangular cookie crumble topping is of 35.1 m².
What is the surface area of a rectangular prism?The surface area of a rectangular prism of height h, width w and length l is given by:
S = 2(hw + lw + lh).
For this problem, the bottom is not covered, hence the surface area will be of:
S = 2(hw + lh) + lw.
The dimensions are given as follows:
l = 4.45, w = 4.06, h = 1.
Hence the surface area is:
S = 2(4.06 x 1 + 4.45 x 1) + 4.45 x 4.06 = 35.1 m².
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A food and nutrition organization reported that 58 out of every 100 people want to lose weight. If 499 students were surveyed at random in the student union,
how many would want to lose weight? Round to the nearest whole number.
About
of the students would want to lose weight.
Answer:
500
Step-by-step explanation:
At this year's State Fair, there was a dice rolling game. If you rolled two dice and got a sum of 2 or 12, you won $25. If you rolled a 7, you won $2. Any other roll was a loss. It cost $1 to play one game with one roll of the dice. What is the expectation of the game? Round your answer to the nearest cent. The expectation of this game is approximately dollars. X
The probability to win 25$ and 2$ are 1/18 and 1/6
Probability is the ratio of favorable cases to total sample cases.
If two dice are rolled,
Total sample cases or combinations= 6x6=36
The favorable cases to get 2 or 12 ={1,2}{6,6}= 2 total cases
The favorable cases to get 7 ={1,6}{6,1}{2,5}{5,2}{3,4}{4,3}=6 cases
The probability to win 25$=2/36=1/18
The probability to win 2$=6/36=1/6
Thus the probability to win 25$ and 2$ are 1/18 and 1/6
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Main Answer: $1.27
Given data:
If you rolled two dice and got a sum of 2 or 12, you won $25. If you rolled a 7, you won $2. Any other roll was a loss. It cost $1 to play one game with one roll of the dice.
Solving part
Step-by-step explanation:
For 2 and 12
Number of possible outcomes = 2
Gain = 25 -1 = $24
For 7
Number of possible outcomes = 6
Gain = 5-1 = $4
For others
Number of possible outcomes = 36-(4+6) = 26
Gain = -$1
Expectation of the gain;
X= (2×$24 + 6×$4 - 26×1)/36
X = $46/36
X = $1.27
Final Answer: $1.27
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TIME REMAINING
48:03
The price of a pair of jeans was $31, but due to an increase the price went up to 129% of what it was originally.
Which is the best estimate of the current price of the pair of jeans?
$36
$39
$48
$52
The best estimation of the current of the jeans is $39.
What is percentage?Percentage is the value per hundredth.Fractions and decimas can be converted into percentage by multiplying them with 100.
According to the given question a pair of jeans was 31 dollars but due to an increase the price went up to 129% of what it was originally.
Let us assume that the new price of the jeans be x dollars.
So, 129% of 31 which is (129/100) is the new price.
∴ (129/100)×31 = x
1.29×31 = x.
x = 39.99 dollars.
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