Answers
This is a telescoping sum. The K-th partial sum is
[tex]S_K = \displaystyle \sum_{k=1}^K \left(\frac1{\sqrt{k+1}} - \frac1{\sqrt{k+3}}\right) \\\\ ~~~= \left(\frac1{\sqrt2} - \frac1{\sqrt4}\right) + \left(\frac1{\sqrt3} - \frac1{\sqrt5}\right) + \left(\frac1{\sqrt4} - \frac1{\sqrt6}\right) + \left(\frac1{\sqrt5} - \frac1{\sqrt7}\right) + \cdots \\\\ ~~~~~~~~+ \left(\frac1{\sqrt{K-1}} - \frac1{\sqrt{K+1}}\right) \\\\ ~~~~~~~~+ \left(\frac1{\sqrt K} - \frac1{\sqrt{K+2}}\right) + \left(\frac1{\sqrt{K+1}} - \frac1{\sqrt{K+3}}\right)[/tex]
[tex]\displaystyle = \frac1{\sqrt2} + \frac1{\sqrt3} - \frac1{\sqrt{K+2}} - \frac1{\sqrt{K+3}}[/tex]
As [tex]K\to\infty[/tex], the two trailing terms will converge to 0, and the overall infinite sum will converge to
[tex]\displaystyle \sum_{k=1}^\infty \left(\frac1{\sqrt{k+1}} - \frac1{\sqrt{k+3}}\right) = \lim_{k\to\infty} S_k = \boxed{\frac1{\sqrt2} + \frac1{\sqrt3}}[/tex]
By the limit comparison test, the expression √[1 / (1 + 1 / k)] - √[1 / (1 + 3 / k)] has a limit, then the expression [1 / √(k + 1)] / [1 /√k] - [1 / √(k + 3)] / [1 /√k] has a limit and the series ∑ [1 / √(k + 1)] - ∑ [1 / √(k + 3)] is convergent.
Is the series convergent?
Herein we have a series that involves radical components. First, we simplify the expression given:
∑ [1 / √(k + 1) - 1 / √(k + 3)] = ∑ [1 / √(k + 1)] - ∑ [1 / √(k + 3)]
The convergence of the series can be proved by the limit comparison test, where each component of the subtraction of the series is compared with a series that is convergent. We notice that both 1 / √(k + 1) and 1 / √(k + 3) resembles the expresion 1 /√k. Then, we have the following subtraction of ratios:
[1 / √(k + 1)] / [1 /√k] - [1 / √(k + 3)] / [1 /√k]
√k / √(k + 1) - √k / √(k + 3)
√[k / (k + 1)] - √[k / (k + 3)]
Then, by using the limit property for rational functions we find the following result for n → + ∞:
√[1 / (1 + 0)] - √[1 / (1 + 0)]
√1 - √1
1 - 1
0
By the limit comparison test, the expression √[1 / (1 + 1 / k)] - √[1 / (1 + 3 / k)] has a limit, then the expression [1 / √(k + 1)] / [1 /√k] - [1 / √(k + 3)] / [1 /√k] has a limit and the series ∑ [1 / √(k + 1)] - ∑ [1 / √(k + 3)] is convergent.
Remark
The statement is incomplete and complete form cannot be found, therefore, we decided to determine if the series is convergent or not.
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Related Questions
The scale of a map is 1:250000. On the map a large forest has an area of 6cm². Calculate the actual area of the forest. Give your answers in square kilometers.
Answers
The distance on map exists 32 cm and actual area exists 37.5 km².
How to estimate the actual area of the forest in square kilometers?
Given: Scale of the map exists at 1:250000.
(a) Distance between two cities = 80 km
= 80000 m
= 8000000 cm
Distance on map = 8000000 [tex]*[/tex] 1/ 250000
= 32 cm
(b) Area of map = 6 cm²
Actual area = [tex]6(250000)^2[/tex] cm²
[tex]= 6 * 625 * 10^8[/tex] cm²
[tex]= 3750 * 10^8/ 10^{10}[/tex]
= 37.5 km²
Therefore, the distance on map exists 32 cm and actual area exists 37.5 km².
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What transformation of the parent function f(x) is made to get 4f(x+2)?
A. a vertical stretch by 4 and 2 units in the x-direction
B. a vertical stretch by 4 and horizontal shrink by 1/2
C. a vertical stretch by 4 and -2 units in the x-direction
D. a vertical stretch by 4 and horizontal shrink by 2
Answers
The correct transformation of f(x) to get 4f(x+2) is by a vertical shift by 4 and 2 units in the x direction.
Given a function f(x) and other function 4f(x+2).
We are required to choose a transformation that will give 4f(x+2) from a function f(x).
Function is like a relationship between two or more variables that are expressed in equal to form. The values which we enter in a function is known as domain and the value that we get as a result are known as range.
So, we have to reach 4f(x+2) from f(x).
The value of function is usually expressed on y axis and variable x on x axis.
To increase the value of x by 2 we have to stretch it in x direction.To find the 4 times value of function we have to stretch the y axis to 4 units.
Hence the correct transformation of f(x) to get 4f(x+2) is by a vertical shift by 4 and 2 units in the x direction.
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the product of 1540 and m is a square number. find the smallest possible value of m
Answers
The smallest possible value of m according to the task is; 1/1540.
What is the smallest possible value of m?Since it follows from the task content that the product of 1540 and m is a square number and the smallest possible small number is; 1.
The equation which holds true is; 1540 × m = 1
Consequently, m = 1/1540.
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i need help i cant find the answer to this question
The Bobcats football coach logged the following yardage gains and losses over four plays of a game.
Gain 25x yards.
Gain 0.9y yards.
Lose 12y yards.
Lose 5.2x yards.
What is the net yardage for these four plays?
Enter your answer as an expression, like this: 42x+53y
Answers
Answer:
Lose 12y yards.
Lose 5.2x yards.
What is the net yardage for these four plays?
is the que correct
Answer:
19.8x - 11.1y
Step-by-step explanation:
Given information:
Gain 25x yardsGain 0.9y yardsLose 12y yardsLose 5.2x yards"Gain" means to add the value.
"Lose" means to subtract the value.
Therefore, net yardage is:
⇒ 25x + 0.9y - 12y - 5.2x
Collect like terms:
⇒ 25x - 5.2x + 0.9y - 12y
Factor out the common variables x and y:
⇒ x(25 - 5.2) + y(0.9 - 12)
Carry out the operations in the parentheses:
⇒ x(19.8) + y(-11.1)
Therefore, the net yardage is 19.8x - 11.1y
Laura completed the following steps to find a product.
Multiply: Three-sevenths times 8
Step 1: 8 times three-sevenths
Step 2: StartFraction (8 plus 3) over 7 EndFraction
Step 3: Eleven-sevenths
Step 4: 1 and four-sevenths
In which step did Laura make her first mistake?
Step 1
Step 2
Step 3
Step 4
Answers
The solution to the product of 8 and three-seventh is 2 2/7. According to Laura, she made mistake in step 2 by adding 8 and 3 instead of multiplying
Multiplication of fractions and integers
Fractions are written as a ratio of two integers. For instance a/b is a fraction where a and b are integers.
Given the following equation
Multiply 3/7 and 8
This is expressed mathematically as;
3/7 * 8
Step 1: Swap to have;
8 * 3/7
Step 2: Group the numerator
(8*2)/7
Step 3; Simplify
16/7
Step 4; Convert to mixed fraction
16/7= 2 2/7
The solution to the product of 8 and three-seventh is 2 2/7. According to Laura, she made mistake in step 2 by adding 8 and 3 instead of multiplying
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Can someone please help me with this?
Answers
Answer: [tex]\Large\boxed{f(-9)=-189}[/tex]
Step-by-step explanation:
f(x) = -3x² - 6x
Requirements of the question
Find the value of f(-9)
Substitute values into the given function
f(x) = -3x² - 6x
f(9) = -3 (-9)² - 6 (-9)
Simplify the exponent
f(9) = -3 (81) - 6 (-9)
Simplify by multiplication
f(-9) = (-243) - (-54)
Simplify by subtraction
[tex]\Large\boxed{f(-9)=-189}[/tex]
Hope this helps!! :)
Please let me know if you have any questions
2
5. One flight took a total of 4.6 hours. Write this number as a mixed number
and as an improper fraction. Show your work in the space below. Remember
to check your solution.
Answers
Step-by-step explanation:
4.6 hrs is 46over 10 which is
13 over 5 or 2 and 3over 5
Which is the greatest fraction in this list? 1/2,3/4,5/8,11/16
Answers
Answer:
3/4
Step-by-step explanation:
multiply each fraction by 100 or you could find the LCM
The total cost of gasoline varies directly with the number of gallons purchased. kathy pays $23.36 for 16 gallons of gasoline. which equation shows the relationship between the total cost of gasoline, c, and the number of gallons purchased, n?
Answers
Equation (A) C= 1.46n shows the relationship between the total cost of gasoline, c, and the number of gallons purchased, n.
What is an equation?An equation is a formula in mathematics that expresses the equality of two expressions by connecting them with the equals sign =. The word equation and its cognates in various languages may have somewhat different definitions; for example, in French, an équation is defined as including one or more variables, whereas in English, an equation is any well-formed formula consisting of two expressions linked by an equals sign.To find the right equation:
In order to find the constant rate, we would divide 23.36/16 which gives you 1.46. That is the price of 1 gallon which would change depending on the amount of falling a purchased (n) and give you the total price of (C).So, C=1.46nTherefore, equation (A) C= 1.46n shows the relationship between the total cost of gasoline, c, and the number of gallons purchased, n.
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The complete question is given below:
The total cost of gasoline varies directly with the number of gallons purchased. Kathy pays $23.36 for 16 gallons of gasoline. Which equation
shows the relationship between the total cost of gasoline, c, and the number of gallons purchased, n?
A.C= 1.46n
B. n = 1.46c
C.C = 23.361
D. n = 23.36c
How many solutions does the equation 4y − 4y − 12 = 14 − 2 have? one zero two many
Answers
Answer:Many
Step-by-step explanation:
As the unknown cancels itself out, it could be absolutely any number. So there are many solutions.
How many ways can a president, vice-president, secretary, and treasurer be chosen from a club with 9 members?
Answers
Answer:
3024
Step-by-step explanation:
Each chosen member (out of 9 members) will occupy a different position
out of these four (president , vice-president , secretary , treasurer).
So, here we have to calculate the Number of Permutations of 9 members Taken 4 at a Time :
= 9P4
= 9 × 8 × 7 × 6
= 3024
It takes a machine at a seafood company 20 s to clean 3 1 ib of shrimp _ 3
Answers
It takes a machine at a seafood company 20 seconds to clean 3 pounds of shrimp. The rate of the machine is 0.15 pound per second
What is an equation?An equation is an expression that shows the relationship between two numbers and variables.
An independent variable is a variable that does not depend on any other variable for its value whereas a dependent variable is a variable that depend on any other variable for its value.
It takes a machine at a seafood company 20 seconds to clean 3 pounds of shrimp. Hence:
Rate of the machine = 3 pounds / 20 seconds = 0.15 pound per second
It takes a machine at a seafood company 20 seconds to clean 3 pounds of shrimp. The rate of the machine is 0.15 pound per second
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Rhett is solving the quadratic equation 0= x2 – 2x – 3 using the quadratic formula. which shows the correct substitution of the values a, b, and c into the quadratic formula? quadratic formula: x = startfraction negative b plus or minus startroot b squared minus 4 a c endroot over 2 a endfraction
Answers
Answer:
Step-by-step explanation:
0= x2 – 2x – 3
a = 1, b = -2 and c = -3.
Answer:
A on edge
Step-by-step explanation:
Give me the brainiest please
A 75-gallon tank is filled with brine (water nearly saturated with salt; used as a preservative) holding 11 pounds of salt in solution. A salt solution containing 0.6 pounds of salt per gallon is added to the tank at the rate of 3gal/min. The contents of the tank are continuously and thoroughly mixed and drained out at thirteen quarts per minute. What is the amount of salt in the tank after an hour
Answers
Let [tex]A(t)[/tex] = amount of salt (in pounds) in the tank at time [tex]t[/tex] (in minutes). Then [tex]A(0) = 11[/tex].
Salt flows in at a rate
[tex]\left(0.6\dfrac{\rm lb}{\rm gal}\right) \left(3\dfrac{\rm gal}{\rm min}\right) = \dfrac95 \dfrac{\rm lb}{\rm min}[/tex]
and flows out at a rate
[tex]\left(\dfrac{A(t)\,\rm lb}{75\,\rm gal + \left(3\frac{\rm gal}{\rm min} - 3.25\frac{\rm gal}{\rm min}\right)t}\right) \left(3.25\dfrac{\rm gal}{\rm min}\right) = \dfrac{13A(t)}{300-t} \dfrac{\rm lb}{\rm min}[/tex]
where 4 quarts = 1 gallon so 13 quarts = 3.25 gallon.
Then the net rate of salt flow is given by the differential equation
[tex]\dfrac{dA}{dt} = \dfrac95 - \dfrac{13A}{300-t}[/tex]
which I'll solve with the integrating factor method.
[tex]\dfrac{dA}{dt} + \dfrac{13}{300-t} A = \dfrac95[/tex]
[tex]-\dfrac1{(300-t)^{13}} \dfrac{dA}{dt} - \dfrac{13}{(300-t)^{14}} A = -\dfrac9{5(300-t)^{13}}[/tex]
[tex]\dfrac d{dt} \left(-\dfrac1{(300-t)^{13}} A\right) = -\dfrac9{5(300-t)^{13}}[/tex]
Integrate both sides. By the fundamental theorem of calculus,
[tex]\displaystyle -\dfrac1{(300-t)^{13}} A = -\dfrac1{(300-t)^{13}} A\bigg|_{t=0} - \frac95 \int_0^t \frac{du}{(300-u)^{13}} [/tex]
[tex]\displaystyle -\dfrac1{(300-t)^{13}} A = -\dfrac{11}{300^{13}} - \frac95 \times \dfrac1{12} \left(\frac1{(300-t)^{12}} - \frac1{300^{12}}\right) [/tex]
[tex]\displaystyle -\dfrac1{(300-t)^{13}} A = \dfrac{34}{300^{13}} - \frac3{20}\frac1{(300-t)^{12}}[/tex]
[tex]\displaystyle A = \frac3{20} (300-t) - \dfrac{34}{300^{13}}(300-t)^{13}[/tex]
[tex]\displaystyle A = 45 \left(1 - \frac t{300}\right) - 34 \left(1 - \frac t{300}\right)^{13}[/tex]
After 1 hour = 60 minutes, the tank will contain
[tex]A(60) = 45 \left(1 - \dfrac {60}{300}\right) - 34 \left(1 - \dfrac {60}{300}\right)^{13} = 45\left(\dfrac45\right) - 34 \left(\dfrac45\right)^{13} \approx 34.131[/tex]
pounds of salt.
PLEASE HELP!! i’ll give brainliest
A. ||
B.
C. neither they are skew lines
Answers
Answer:
I think it's A (parallel)
Answer:
A. its parallel
Step-by-step explanation:
all angles are equal... two sides are perpendicular and two are parallel
Solve the following system using the algebraic method of substitution. Verify your solution.
x + 2y = -5
3x - y = -1
Solve the following linear system using the algebraic method of elimination. Verify your solution.
x + 2y = 2
3x + 5y = 4
Solve the following linear system algebraically. State why you chose the method you used.
x + 3y = 7
2x + 4y = 11
Answers
Answer: Look into step by step
Step-by-step explanation:
1. Multiply the second equation by 2; 6x - 2y = -2 then add to first equation
7x = -7 so x = -1, substitute x = -1 into the first equation -1 + 2y = -5 so y = -2
2. Multiply the first equation by 3; 3x + 6y = 6 then subtract it by second equation
y = 2, substitute y = 2 into first equation, x + 4 = 2, x = -2
3. Multiply the first equation by 2; 2x + 6y = 14 then subtract it by second equation
2y = 3 so y = 1.5, substitute y = 1.5 into the first equation x + 4.5 = 7, x = 2.5
I chose this method as it is easy
Phillippe wants to drink at least 72 ounces of water every day while at work. He works 6 hours a day. How many ounces of water must Phillippe drink each hour? Write and solve an inequality to answer the question.
Answers
Answer:
the answer to this question is 12
Step-by-step explanation:
72÷6=12
What is the twentieth term in the sequence 23, 43, 63, 83 …?
Answers
Answer:
403 (hope this helps)
Step-by-step explanation:
graph the equation by translating y=|x-1|
i need help like super fast-
Answers
Answer:
Graph b
Step-by-step explanation:
y = |x| is a "v"at x = 0
y = |x-1| is a "v" shifted to the right by 1 unit
5) For the fraction 3/25, (a) write a percent and (b) write a decimal.
Answers
Find the dimensions of a rectangle with an area of 2x^2-7x-10
Answers
Answer:
Step-by-step explanation:
I gotcha my dude.......
x-5 and 2x+3
Identify the standard form of the equation by completing the square.
4x2 − 9y2 − 8x + 36y − 68 = 0
Answers
Answer:
[tex]\dfrac{(x-1)^2}{9}-\dfrac{(y-2)^2}{4}=1[/tex]
Step-by-step explanation:
Given equation:
[tex]4x^2-9y^2-8x+36y-68=0[/tex]
This is an equation for a horizontal hyperbola.
To complete the square for a hyperbola
Arrange the equation so all the terms with variables are on the left side and the constant is on the right side.
[tex]\implies 4x^2-8x-9y^2+36y=68[/tex]
Factor out the coefficient of the x² term and the y² term.
[tex]\implies 4(x^2-2x)-9(y^2-4y)=68[/tex]
Add the square of half the coefficient of x and y inside the parentheses of the left side, and add the distributed values to the right side:
[tex]\implies 4\left(x^2-2x+\left(\dfrac{-2}{2}\right)^2\right)-9\left(y^2-4y+\left(\dfrac{-4}{2}\right)^2\right)=68+4\left(\dfrac{-2}{2}\right)^2-9\left(\dfrac{-4}{2}\right)^2[/tex]
[tex]\implies 4\left(x^2-2x+1\right)-9\left(y^2-4y+4\right)=36[/tex]
Factor the two perfect trinomials on the left side:
[tex]\implies 4(x-1)^2-9(y-2)^2=36[/tex]
Divide both sides by the number of the right side so the right side equals 1:
[tex]\implies \dfrac{4(x-1)^2}{36}-\dfrac{9(y-2)^2}{36}=\dfrac{36}{36}[/tex]
Simplify:
[tex]\implies \dfrac{(x-1)^2}{9}-\dfrac{(y-2)^2}{4}=1[/tex]
Therefore, this is the standard equation for a horizontal hyperbola with:
center = (1, 2)vertices = (-2, 2) and (4, 2)co-vertices = (1, 0) and (1, 4)[tex]\textsf{Asymptotes}: \quad y = -\dfrac{2}{3}x+\dfrac{8}{3} \textsf{ and }y=\dfrac{2}{3}x+\dfrac{4}{3}[/tex][tex]\textsf{Foci}: \quad (1-\sqrt{13}, 2) \textsf{ and }(1+\sqrt{13}, 2)[/tex]find the equation of a line that passes through the point (4, 3) and is parallel to the line 2x-2y=11
Answers
Answer:
The slope of the line would be y=x-1
Step-by-step explanation:
y=x-1
Subsite
3=4-1
3=3 True,
Thus y=x-1
What are the parts of an algebraic expression, and how do they relate to polynomial expressions?
Answers
The parts of algebraic expressions related to polynomials are variables and coefficients.
What are the parts of algebraic expressions?The parts of algebraic expressions are;
Variables which are letters that represent numbersCoefficients are numbers that multiply the variables Constant is a number that is not multiplied by any variablePolynomials are algebraic expressions composed of variables, constants and exponents, that are combined using the mathematical operations such as addition, subtraction, and multiplication
Polynomials consists of variables and coefficients. The variables in polynomials are also called indeterminates. The coefficients also multiply this variables.
Thus, the parts of algebraic expressions related to polynomials are variables and coefficients.
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. What is the midpoint of a line segment connecting the points (−4,6) and (8,2)?
Answers
if in Ashford the lights come on at 6:20pm and go off 11 1/2 hours later what time will it be?
Answers
Because 6:20 plus 1/2 is 6:50 because 1/2 of an hour is 30 min then 6:50 plus 11 would 17:50-12=5.50am
A parent made x cupcakes for each of the 109 students in the fourth grade. Which expression could be used to determine the total number of cupcakes made?
A. x/109
B. 109/x
C.109x
D.109+x
Answers
Answer:
C. 109x
Step-by-step explanation:
We know that there are 109 students in the fourth grade, who each have x cupcakes. In order to find the total amount of cupcakes, we could add all of the cupcakes together to get:
x + x + x + x + .... x
109 times because there are 109 students.
This can be simplified to become 109x.
If y = -9 when x = 3, find y when x = -6. when x = -5.
Answers
Answer:
When x is -6, y will be 18
When x is -5 y will be 15
Step-by-step explanation:
Create a proportion
y/x
-9/3 = x/-6 What do we need to multiply 3 by to get 6? -2, so we will multiply -9 by -2 to get 18
Some people like to solve by cross multiplying.
That would give us
(-9)(-6) = 3x
54 = 3x divide both sides by 3
18 = x
Set up the second problem as a proportion
-9/3 = x/-5 This is a little trickier. I am trying to think what I have to multiply 3 by to get -5. 3(-5/3) will give me -5, so I will multiply the top number that I know (-9) by (-5/3) to find my x.
(-9)(-5/3) = 45/3 or 15
Situation:
A student in Greece discovers a pottery
bowl that contains 65% of its original
amount of C-14.
N = Noe-kt
No inital amount of C-14 (at time
t = 0)
N = amount of C-14 at time t
k = 0.0001
t = time, in years
Find the age of the pottery bowl to the nearest
year.
Enter the correct answer.
000
?
DONE
Answers
The age of the pottery bowl to the nearest year is 4307 years. Using exponential decay or growth formula, the required value is calculated.
What is the formula for exponential growth/decay?The formula for the exponential growth/decay is
[tex]N=N_0e^-^k^t[/tex]
Where,
N - the total amount after time t
N₀ - the initial amount
k - growth or decay rate
t - time
Calculation:It is given that,
A student in Greece discovers a pottery bowl that contains 65% of its original amount of C-14.
⇒ N(t) = 0.65N₀
But we have k = 0.0001.
Then,
[tex]N(t)=N_0e^-^k^t[/tex]
0.65 N₀ = N₀ [tex]e^{-(0.0001)t}[/tex]
⇒ 0.65 = [tex]e^{-(0.0001)t}[/tex]
⇒ ln [tex]e^{-(0.0001)t}[/tex] = ln 0.65
⇒ -(0.0001)t = -0.4307
⇒ t = 0.4037/0.0001
∴ t = 4307 years.
Thus, the age of the pottery bowl is 4307 years.
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Show do i solve for the missing answers
Answers
g(x) + f(x)
x - 1 + x^2 + 4
x^2 + x + 3
(g - h)(x)
g(x) - h(x)
3x + 1 - (2x - 2)
3x + 1 - 2x + 2
x + 3