Using the exp-log technique,
[tex]\displaystyle \lim_{n\to\infty} \frac{\left(1 \times 2^2 \times 3^3 \times \cdots \times n^n\right)^{\frac1{n^2}}}{n^{\frac{n+1}{2n}}}[/tex]
[tex]\displaystyle = \exp\left(\lim_{n\to\infty} \frac{\ln\left(\prod\limits_{k=1}^n k^k\right)}{n^2} - \frac{n+1}{2n} \ln(n)\right)\right)[/tex]
[tex]\displaystyle = \exp\left(\lim_{n\to\infty} \frac{\sum\limits_{k=1}^n k\ln(k)}{n^2} - \frac{n+1}{2n} \ln(n)\right)\right)[/tex]
We have ln(k) = ln(k/n) + ln(n), and so we can rewrite the limand as
[tex]\displaystyle = \exp\left(\lim_{n\to\infty} \left(\frac1n \sum\limits_{k=1}^n \frac kn \ln\left(\frac kn\right) + \frac{\ln(n)}{n^2} \sum_{k=1}^n 1 - \frac{n+1}{2n} \ln(n)\right)\right)[/tex]
Then the first sum converges to a definite integral,
[tex]\displaystyle \lim_{n\to\infty} \frac1n \sum_{k=1}^n \frac kn \ln\left(\frac kn\right) = \int_0^1 x \ln(x) \, dx = -\frac14[/tex]
while the remaining terms vanish since
[tex]\displaystyle \frac{\ln(n)}{n^2} \sum_{k=1}^n 1 = \frac{\ln(n)}{n^2} \times \frac{n(n+1)}2 = \frac{n+1}{2n}\ln(n)[/tex]
So the limit is [tex]\boxed{e^{-1/4}}[/tex].
Since
[tex]\dfrac1{x(x+1)} = \dfrac1x - \dfrac1{x+1}[/tex]
it's easy to show that the integral reduces to
[tex]\displaystyle \int_k^{k+1} \frac{k+1}{x(x+1)} \, dx = (k+1) \ln\left(\dfrac{(k+1)^2}{k(k+2)}\right)[/tex]
so we can write the sum as
[tex]I = \displaystyle \sum_{k=1}^{98} \int_k^{k+1} \frac{k+1}{x(x+1)} \, dx = \sum_{k=2}^{99} k \ln\left(\frac{k^2}{k^2-1}\right)[/tex]
We have k²/(k² - 1) > 1 for all k, so that ln(k²/(k² - 1)) > ln(1) = 0. We can see the first 3 terms of the sum already exceed 1 > 49/50, so (D) is true.
Numerical computation of the sum suggests I < ln(99), but I have yet to come up with an analytical solution for this bound...
Please help
What are all the possible
outcomes that add up to
7 when you roll two
dice? Include 3+4 and
4+3.
Please help
Answer:
5+2, 2+5, 4+3, 3+4, 6+1, 1+6, 7+0, 0+7
Step-by-step explanation:
Hope this helps!
The grid represents all of the following decimal numbers except
0.300
0.3
0.30
0.030
Answer:
0.030
Step-by-step explanation:
0.030 is 3/100
what is show above is 3/10 and 0.300, 0.3, 0.30 are all equal to 3/10
V
Graph the line y= -7.
Answer:
There is no slope, so we draw a line parallel to the x-axis and the width of the origin is equal to -7
Find the square of 75.
Answer:
8.66025 or 5√3
Step-by-step explanation:
Factor
25
out of
75
.
√
25
(
3
)
Rewrite
25
as
5
2
.
√
5
2
⋅
3
Pull terms out from under the radical.
5
√
3
The result can be shown in multiple forms.
Exact Form:
5
√
3
Decimal Form:
8.66025403
Find the linear function given h(4)=-13 h(-2)=17 find h(1)
Answer:
the linear function: y=-5x + 7
h(1)=2
Find the slope and the y-intercept of the graph of the linear equation.
Explanation
The given equation y = 2x+3 is in the form y = mx+b
Compare the terms to see that 2x matches with mx, so m = 2
The other term matches with b, so b = 3
m is the slope and b is the y intercept, so that's how I got those answers above.
A stadium can seat 46,000 people for a baseball game. One day, 49,442 people attended a game. How many people had to stand because they did not have a seat?
Step-by-step explanation:
49,442-46,000=3,442
Which value of x makes this equation true?
HELPPP
Answer:
x=-2
Step-by-step explanation:
First you remove the parenthesis by distributing.
Second you collect like terms
Third move the terms to where they can be added
Fourth calculate
Fifth divide
Then boom answer (you got helped by an 8th grader)
Write the ratio: 0.28 to 0.84 as a fraction in lowest terms.
What is the surface area of a box with a length of 6 ft, a width of 3 ft, and a height of 2½ ft?
Answer:
Step-by-step explanation:
Remark
The surface area is the area of all six sides when the object is a box.
Formula
SA = 2*L*w + 2*L*h + 2*w*h
Givens
L = 6 ft
w= 3 ft
h = 2.5 feet
Substitute and Solution
SA = 2*6*3 + 2*6*2.5 + 2*3*2.5
SA = 36 + 30 + 15
SA = 81
(-9.206) x(14.04)=
in a decimal
Answer:
=−129.25224x
Step-by-step explanation:
(−9.206x)(14.04) you just simplify
I’m like really bad at math so can someone please help me pleaseee
Answer:
The biggest differentiating factor between a function and NOT a function
is that a function gives only one output for any given input. so no matter how many times you use the same value in the function, you will get the same output
A relation which is not a function will have 2 or more outputs for a single input
So if we have more than 1 similar input values in the table and given them different outputs, the relation will NOT be a function
I will take 2 similar inputs "1" and give them different outputs, the rest of the table can be randomly filled
x y
1 4
3 7
5 9
1 2
For what value of X must ABCD be a parallelogram?
Answer:
x = 9
Step-by-step explanation:
If ABCD is a parallelogram, then 6x = 7x - 9
as the point of intersection of the 2 diagonals is the midpoint of AC
6x = 7x - 9
Subtract 6x from both sides:
⇒ 0 = x - 9
Add 9 to both sides:
⇒ 9 = x
Solution:
We know that:
[tex]\frac{AC}{2} = 6x[/tex]This means that:
[tex]AC = (6x)(2) = 12x[/tex][tex]AC = 12x = 6x + 7x - 9[/tex]Step-by step calculations:
Simplify the RHS:
[tex]12x = 6x + 7x - 9[/tex]=> [tex]12x = 13x - 9[/tex]Subtract 13x both sides:
=> [tex]-13x + 12x = 13x -13x - 9[/tex]=> [tex]-x = -9[/tex]Divide -1 both sides:
=> [tex]-x = -9[/tex]=> [tex]\frac{-x}{-1} = \frac{-9}{-1}[/tex]=> [tex]x = 9[/tex]Graph the solution of this inequality 3/4(x+8)>1/2(2x+10)
Answer:
I'm terrible at explaining so here's a screenshot
- Ripper
Step-by-step explanation:
Can you answer this one please
Answer:
50⁰
Step-by-step explanation:
Since j and k are parallel lines cut by two transversals,
2+3+5=180⁰ (Allied angles)
We are given 2(60⁰) and 5(70⁰)
so,
60⁰+3+70⁰=180⁰
60+70=130
130⁰+3=180⁰
Bring the 130⁰ to the R.H.S
3=180⁰-130⁰
Therefore,
3=50⁰
Rate of Change Find the rate of change given 2 points 1. (9, 4)(7,8) 2. (0,5) (5, 15) (-8, -2)(-3, 10) (5, 12)(5, 12) 5. (23, 43)(56, 17) 3. A.
Answer:
(y₂ - y₁) / (x₂ - x₁)
Step-by-step explanation:
35. Find the perimeter and area of the following figure.
10
3
6
5
(The measures are given in centimeters)
Brainliest If correct help
Answer:
I already answered before but i hope u see this 1, the 1st option
Step-by-step explanation:
PLZ do it. hope it helps :)
is 5/10 equivalent to 1/6
Answer:
No, its not equivalent
1/2 is equivalent though
Please mark as brainliest
Answer:
No
Step-by-step explanation:
Given the following question:
[tex]\frac{5}{10}[/tex]
You can prove 5/10 isn't equivalent to 1/6 by simply reducing 5/10.
[tex]\frac{5}{10}[/tex]
[tex]\frac{5}{10}\div5=\frac{1}{2}[/tex]
[tex]=\frac{1}{2}[/tex]
[tex]\frac{1}{2} > \frac{1}{6}[/tex]
5/10 isn't equivalent to 1/6, so your answer is "no."
Hope this helps.
Solve |y + 2| > 6
thanks
Answer:
[tex]\huge\boxed{y>4\ \vee\ y<-8}\\\boxed{y\in(-\infty;\ -8)\ \cup\ (4;\ \infty)}[/tex]
Step-by-step explanation:
[tex]|y+2|>6\iff y+2>6\ \vee\ y+2<-6\qquad|\text{subtract 2 from both sides}\\\\y+2-2>6-2\ \vee\ y+2-2<-6-2\\\\y>4\ \vee\ y<-8\Rightarrow y\in(-\infty;\ -8)\ \cup\ (4;\ \infty)[/tex]
Jana can spend, at most, $200 on school clothes.
Which inequality represents the amount that Jana can spend on clothes?
Answer:
I believe the answer is c
Enter the improper fraction as a mixed number.
23/9
Answer:
2 5/9 is the mixed number
Step-by-step explanation:
2x9=18
23-18=5
Answer:
2 5/9
Step-by-step explanation:
Just divide by 9 and leave the remainder over 9
So 23/9 is 2 R 5
So the answer is 2 5/9
helppppppppppppppppppppppppppppppppppppppp
Answer:
20
Step-by-step explanation:
If you turn it into a rectangle, then you can see one side is 4 squares long, and the other is 5 squares long. 4 x 5= 20
Which of the following is FALSE?
A. A midpoint divides a segment into two equal parts.
B. An angle bisector divides an angle into two equal parts.
C. All right angles are congruent.
D. Vertical angles are acute angles?
Answer:
D
Step-by-step explanation:
Vertical angles are just angles formed by intersecting lines, there is no rule that states they must be acute (they can also be right or obtuse)
The option that is False is that Vertical angles are acute angles.
What is the meaning of the term congruent?In mathematics, the term congruent means a figure, shape, or angle that has two equal parts, size, and length.
It is true that the midpoint of a segment refers to the line that splits the segment into two congruent(equal parts)Also, at any two equidistant points, an angle bisector divides an angle into two equal parts.All right angles are also congruent to one another.However, vertical angles assume a right angle and they are perpendicular to the straight line which makes them 90°, but we know that acute angles are angles that are less than 90°.
Therefore, we can conclude that vertical angles are not acute angles.
Learn more about congruent angles here:
https://brainly.com/question/24281994
#SPJ5
Fatih parasının ilk %30unu sonra %35ini harcar ve geriye 105 TL’si kalır. Fatih’in toplamda kaç TL’si vardır?
Answer:
is good by his guan yes is help to me
A triangle on a coordinate plane is translated according to the rule T–8, 4(x, y). Which is another way to write this rule?
(x, y) → (x + 4, y – 8)
(x, y) → (x – 4, y – 8)
(x, y) → (x – 8, y + 4)
(x, y) → (x + 8, y – 4)
Answer:
(x, y) → (x – 8, y + 4)
Step-by-step explanation:
Given:
A triangle on a coordinate plane is translated according to the rule T–8, 4(x, y).
To Find:
Which is another way to write this rule?
Solve:
Therefore the answer is (x, y) → (x – 8, y + 4)
!Lenvy~
Answer:
(x, y) → (x – 8, y + 4)
Step-by-step explanation:
3.Paul owes his father $56.78. He borrows $20 more from his father. Write an expression to represent his debt. Helpp plzzz asap
Work out the length of X.
х
8 cm
15 cm
Using the Pythagorean theorem which states that;
a²+b²=c²
or
side (1) ²+side (2) ²=hypotenuse²
15²+8²=x²
225+64=x²
289=x²
Take the square root of both sides to get;
x=√289=17
x=17
Step-by-step explanation:
[tex] \longmapsto \: hypotenuse(h) = x \\ \longmapsto \: perpendicular(p) = 15cm \\ \longmapsto \: base(b) = 8cm \\ using \: pythagoras \: theorem \\ \hookrightarrow {h}^{2} = {p}^{2} + {b}^{2} \\\hookrightarrow {x}^{2} = ({15cm}^{2} + ({8cm})^{2} \\ x = \sqrt{225 {cm}^{2} + 64 {cm}^{2} } \\ \hookrightarrow \: x = \sqrt{289 {cm}^{2} } \\ \hookrightarrow \: x = 17cm [/tex]
does anyone know this
Answer:
x ≤ 9
Step-by-step explanation:
Step 1: Write inequality
x - 9 ≤ 2(9 - x)
Step 2: Solve for x
Distribute 2: x - 9 ≤ 18 - 2xAdd 2x on both sides: 3x - 9 ≤ 18Add 9 to both sides: 3x ≤ 27Divide both sides by 3: x ≤ 9Here we see that any solution less than or equal to 9 would work.
Answer:
x ≤ 9
Step-by-step explanation:
x -9 ≤ 2(9 -x)
x -9 ≤ 18 -2x
+2x +2x
3x -9 ≤ 18
+9 +9
3x ≤ 27
[tex]\frac{3x}{3}[/tex] ≤ [tex]\frac{27}{3}[/tex]
x ≤ 9
Find f '(x) for f of x equals the square root of the quantity sine of 4 times x
Answer:
I'm not exactly sure, sorry.
Step-by-step explanation:
[tex]\sqrt{x} sin(4)*x[/tex]
square root of sin(4) times x
The goal of this equation is to find an x value that will make both sides equal.
I probably shouldn't answer this question since I can't really find a solution to it, but I don't really think its solvable.