Determine whether the sequence
a_n = \frac{1^1}{n^2} + \frac{2^1}{n^2} + \cdots + \frac{n^1}{n^2}converges or diverges. If it converges, find the limit.
Converges(y/n):
Limit (if it exist, blank otherwise) :
Answers
a_n = \(\frac{1^1}{n^2} + \frac{2^1}{n^2} + \cdots + \frac{n^1}{n^2}\)
The sequence a_n converges, and its limit is 1/2.
We need to determine if the sequence a_n converges or diverges, where a_n is given by the sum:
a_n = \frac{1^1}{n^2} + \frac{2^1}{n^2} + \cdots + \frac{n^1}{n^2}
First, let's rewrite a_n as a sum of fractions:
a_n = \frac{1}{n^2} (1 + 2 + \cdots + n)
Now, we know that the sum of the first n integers can be represented by the formula:
1 + 2 + \cdots + n = \frac{n(n + 1)}{2}
So, a_n becomes:
a_n = \frac{1}{n^2} \cdot \frac{n(n + 1)}{2}
Simplify the expression:
a_n = \frac{n(n + 1)}{2n^2}
Now we can find the limit of the sequence a_n as n approaches infinity:
lim (n -> ∞) a_n = lim (n -> ∞) \frac{n(n + 1)}{2n^2}
Using L'Hôpital's rule, since the limit is of the form infinity/infinity:
lim (n -> ∞) a_n = lim (n -> ∞) \frac{2n + 1}{4n}
Applying L'Hôpital's rule again:
lim (n -> ∞) a_n = lim (n -> ∞) \frac{2}{4} = \frac{1}{2}
Since the limit of the sequence a_n exists and is finite, the sequence converges.
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Related Questions
Can someone tell me the answer and explain it?
Answers
Answer:
30 in
Step-by-step explanation:
If the scale drawing has a base length of 10 in whe the real object has a base length of 25 inches, the scale of the drawing is 1:25
(25 divided by 2.5 = 10)
Since you already have the base length you need to find the height of the scale drawing. The height of the real object is 15 in so the scale drawing would be 6 in.
(15/2.5 = 6)
Now use the formula A= b x h/2 to find the area of the scale drawing.
A = 10 x 6/2
A = 60/2
A= 30
(-4 2/3) (1 1/2) as a mixed number in simplest form
Answers
Answer:
-7
Step-by-step explanation:
Exercise 2(1/2) We can describe a parabola with the following formula: y=a ∗
x∗2+b ∗
x+c Write a Python script which prompts the user for the values of a, b, c,x, and y and then tests whether the point (x,y) lies on the parabola or not. Print out this information accordingly. Hint: check for equality on both sides of the above equation (==). Exercise 2(2/2) Example output: Input a float for ' a ': 1 Input a float for ' b ': 0 Input a float for ' c ': 0 Input a float for ' x ': 4 Input a float for ' y ': 16 The point (4,16) lies on the parabola described by the equation: y=1∗ x∗∗2+0∗x+0
Answers
The Python script above prompts the user for the values of a, b, c, x, and y, and then tests whether the point (x, y) lies on the parabola described by the equation y=ax^2+bx+c. If the point lies on the parabola, the script prints out a message stating this. Otherwise, the script prints out a message stating that the point does not lie on the parabola.
The function is_on_parabola() takes in the values of a, b, c, x, and y, and then calculates the value of the parabola at the point (x, y). If the calculated value is equal to y, then the point lies on the parabola. Otherwise, the point does not lie on the parabola.
The main function of the script prompts the user for the values of a, b, c, x, and y, and then calls the function is_on_parabola(). If the point lies on the parabola, the script prints out a message stating this. Otherwise, the script prints out a message stating that the point does not lie on the parabola.
To run the script, you can save it as a Python file and then run it from the command line. For example, if you save the script as parabola.py, you can run it by typing the following command into the command line:
python parabola.py
This will prompt you for the values of a, b, c, x, and y, and then print out a message stating whether or not the point lies on the parabola.
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5(x+2)=5x+10
I need two numbers that can make the equation true because it’s infinite solutions but I don’t know what to do please help
Answers
Answer:
6 and 2222
Step-by-step explanation:
5(x+2)=5x+10
If we expand the left half we get:
5x+10=5x+10
That's why there are infinite solutions. 5x+10 will always equal 5x+10 no matter what x is chosen.
If you need two numbers, I suggest 6 and 2222. No reason: 34567 and 098776655 would also work.
Select all of the true statements about the graph. a. It is not a function. b. It is a function. c. It does not represent a proportional relationship. d. It does represent a proportional relationship.
Answers
The true statements about the given graph are :
(b) It is a function.
(c) It does not represent a proportional relationship.
Given a graph.
A function is a relation from a set A to a set B where the elements in set A only maps to one and only one image in set B. No elements in set A has more than one image in set B.
Here take the set A has the x values and the set B has the y values.
By vertical line test, if we draw a vertical line through any point, the line only touches the graph at one point.
So this is a graph of a function.
Proportional relationships are relationships where the ratios of two y values will be equal.
And they will be linear.
Here the graph is not linear and thus not proportional.
Hence the true statements are b and c.
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Help................
Answers
Answer:
C
Step-by-step explanation:
need help with this one asap
Answers
if you're solving it for R, it's r = 3s
if you're solving for S, it's s = r/3
please help quick!
brainliest!
Answers
Answer:
(3, -2)
Step-by-step explanation:
I attached a picture of the work
hope this helps
Answer:
(3,-2)
Step-by-step explanation:
use m a t h w a y! its free and amazing
2-yard piece of ribbon costs $1.26. What is the price per foot?
Answers
1 yard = 3 feet
3ft * 2 yards = 6 feet
$1.26 / 6ft = $0.21 / ft
John has 48 square centimeter tiles he wants to use to create a mosaic. He wants the mosaic to be rectangular with a length that is 2 centimeters longer than the width.
Which equation could John solve to find w, the greatest width in centimeters he can use for the mosaic?
w(w – 2) = 48
w(w + 2) = 48
2w(w – 2) = 48
2w(w + 2) = 48
Answers
The equation that could John solve to find w, the greatest width in centimeters he can use for the mosaic is option B: w(w + 2) = 48.
What is a system of equations?A system of equations is two or more equations that can be solved to get a unique solution. the power of the equation must be in one degree.
The mosaic to be formed is rectangular in shape.
Area of a rectangle = length x width
The length is longer than the width by 2,
So, length = w+2
Area of a rectangle = (w+2) x w
The correct equation is B: w(w + 2) = 48
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Answer: B
Step-by-step explanation:
There are 1,332 people under the age 20 in Pierce City. This represents 14% of the total population. What is the total population
Answers
Answer:
The total population = 9514.2857143 people.
Step-by-step explanation:
Let the total population be represented by X
In the question, we are told that,1332 people are under the age of 20 and this value = 14% of the total population.
Therefore, the total population is calculated as
14% of X = 1332
14/100 × X = 1332
14X/100 = 1332
Cross Multiply
14X = 1332 × 100
X = 1332 × 100/14
X = 9514.2857143
Therefore, the total population = 9514.2857143 people.
Select all that are equivalent to 4/16
1/5
0.25
0.252525
25%
Answers
Answer:
0.25 and 25% are equivalent
Step-by-step explanation:
Translate each of the following word phrases into algebraic expressions using the symbols given.
Eg) Twelve times a number 'x', plus a second number 'y'
= 12x + y
1) Half the product of two numbers x and y, minus five times a third number z.
Answers
Answer:
half that's 1/2
product that's multiplication ×
therefore (x*y)/2- 5z
4 tons of sand cost $6,240.00. What is the price per pound?
Answers
Answer:$0.78
Step-by-step explanation:
use the definitions below to select the statement that is true. a={x∈:xis even}b={x∈:−4
Answers
The true statement is: Set a contains all the elements in set b and more.
The definitions given are:
a = {x ∈ : x is even}
b = {x ∈ : −4 < x ≤ 4}
To find the true statement, we need to compare the two sets.
Looking at set a, it consists of all the even numbers. So, a = {..., -4, -2, 0, 2, 4, ...}
On the other hand, set b consists of all the numbers greater than -4 and less than or equal to 4. So, b = {-4, -3, -2, -1, 0, 1, 2, 3, 4}
Now, let's compare the two sets:
a = {..., -4, -2, 0, 2, 4, ...}
b = {-4, -3, -2, -1, 0, 1, 2, 3, 4}
From the comparison, we can see that every element in set b is also in set a, but set a includes additional elements like {..., -4, ...}.
Therefore, the true statement is: Set a contains all the elements in set b and more.
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2. the butler and the cook have decided to murder their employer. they draw straws to determine which one of them must carry out the dirty deed (so each has the same chance). the butler has four poison tipped pens, two crowbars and four knives and the cook has three rolling pins and seven knives. whoever is chosen to be the murderer will select one of their weapons randomly. a. what is the probability that the murder was committed with a knife? b. given that the murder was committed with a knife, what's the probability that the cook did it?
Answers
a) The probability that the murder was committed with a knife: 0.55
b) The probability that the cook did it, given that the murder was committed with a knife: 0.275
To determine the probability that the murder was committed with a knife, we first need to find the total number of weapons.
The butler has four poison tipped pens, two crowbars and four knives and the cook has three rolling pins and seven knives.
So, the total knives: 4 + 7 = 11
crowbars: 2
poison tipped pens: 4
and rolling pins: 3
So, the total number of weapons: 11 + 2 + 4+ 3 = 20
The probability that the murder was committed with a knife:
p = 11/20
p = 0.55
Now we need to find the probability that the cook did it, given that the murder was committed with a knife.
P = 0.55 × 0.5
P = 0.275
The required probability is 0.275
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What is the expression in factored form?
5x² - 7x-6
O
(5x-2)(x-3)
(x + 3)(5x - 2)
O (5x + 3)(x - 2)
O (5x + 3)(x + 2)
Answers
(5x+3)(x-2)
(5x+3)(x-2)= 5x^2-10x+3x-6= 5x^2-7x-6
given a function f : a → b and subsets w, x ⊆ a, then f (w ∩ x) = f (w)∩ f (x) is false in general. produce a counterexample.
Answers
Therefore, f(w ∩ x) = {0} ≠ f(w) ∩ f(x), which shows that the statement f(w ∩ x) = f(w) ∩ f(x) is false in general.
Let's consider the function f: R -> R defined by f(x) = x^2 and the subsets w = {-1, 0} and x = {0, 1} of the domain R.
f(w) = {1, 0} and f(x) = {0, 1}, so f(w) ∩ f(x) = {0}.
On the other hand, w ∩ x = {0}, and f(w ∩ x) = f({0}) = {0}.
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Ken bought some lollipops. He gave 12 of them to Chin, 3 to Harris, and kept 6 for himself. How many lollipops did Ken buy?
Answers
Students arrive at the Administrative Services Office at an average of one every 12 minutes, and their requests take on average 10 minutes to be processed. The service counter is staffed by only one clerk, Judy Gumshoes, who works eight hours per day. Assume Poisson arrivals and exponential service times. Required: (a) What percentage of time is Judy idle? (Round your answer to 2 decimal places. Omit the "%" sign in your response.) (b) How much time, on average, does a student spend waiting in line? (Round your answer to the nearest whole number.) (c) How long is the (waiting) line on average? (Round your answer to 2 decimal places.) (d) What is the probability that an arriving student (just before entering the Administrative Services Office) will find at least one other student waiting in line? (Round your answer to 3 decimal places.)
Answers
The probability that an arriving student will find at least one other student waiting in line is approximately 0.167.
To solve this problem, we'll use the M/M/1 queueing model with Poisson arrivals and exponential service times. Let's calculate the required values: (a) Percentage of time Judy is idle: The utilization of the system (ρ) is the ratio of the average service time to the average interarrival time. In this case, the average service time is 10 minutes, and the average interarrival time is 12 minutes. Utilization (ρ) = Average service time / Average interarrival time = 10 / 12 = 5/6 ≈ 0.8333
The percentage of time Judy is idle is given by (1 - ρ) multiplied by 100: Idle percentage = (1 - 0.8333) * 100 ≈ 16.67%. Therefore, Judy is idle approximately 16.67% of the time. (b) Average waiting time for a student:
The average waiting time in a queue (Wq) can be calculated using Little's Law: Wq = Lq / λ, where Lq is the average number of customers in the queue and λ is the arrival rate. In this case, λ (arrival rate) = 1 customer per 12 minutes, and Lq can be calculated using the queuing formula: Lq = ρ^2 / (1 - ρ). Plugging in the values: Lq = (5/6)^2 / (1 - 5/6) = 25/6 ≈ 4.17 customers Wq = Lq / λ = 4.17 / (1/12) = 50 minutes. Therefore, on average, a student spends approximately 50 minutes waiting in line.
(c) Average length of the line: The average number of customers in the system (L) can be calculated using Little's Law: L = λ * W, where W is the average time a customer spends in the system. In this case, λ (arrival rate) = 1 customer per 12 minutes, and W can be calculated as W = Wq + 1/μ, where μ is the service rate (1/10 customers per minute). Plugging in the values: W = 50 + 1/ (1/10) = 50 + 10 = 60 minutes. L = λ * W = (1/12) * 60 = 5 customers. Therefore, on average, the line consists of approximately 5 customers.
(d) Probability of finding at least one student waiting in line: The probability that an arriving student finds at least one other student waiting in line is equal to the probability that the system is not empty. The probability that the system is not empty (P0) can be calculated using the formula: P0 = 1 - ρ, where ρ is the utilization. Plugging in the values:
P0 = 1 - 0.8333 ≈ 0.1667. Therefore, the probability that an arriving student will find at least one other student waiting in line is approximately 0.167.
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solve
2-\(\frac{x}{3}\)=6-x
Answers
\(:\implies\sf\:2-\dfrac{x}{3}=6-x\)
Taking LCM in left side, we obtain
\(:\implies\sf\:\dfrac{6-x}{3}=6-x\)
\(:\implies\sf\:6-x=3(6-x)\)
\(:\implies\sf\:6-x=18-3x\)
\(:\implies\sf\:-x+3x=18-6\)
\(:\implies\sf\:2x=12\)
\(:\implies\sf\:x=\dfrac{12}{2}\)
\(:\implies\boxed{\bf{\orange{x=6}}}\)
Suppose that the revenue R, in dollars, from selling x cell phones, in hundreds, is R(x)=−1.9x
2
+256x. The cost C, in dollars, from selling x cell phones, in hundreds, is C(x)=0.05x
3
−3x
2
+55x+200. (a) Find the profit function, P(x)=R(x)−C(x). (b) Find the profit if x=23 hundred cell phones are sold. (c) Interpret P(23). (a) P(x)=−0.05x
3
+1.1x
2
+201x−200 (Use integers or decimals for any numbers in the expression.) (b) P(23)=$ (Type an integer or a decimal. Round to the nearest cent.) (c) Interpret P(23). Choose the correct answer below. A. When 23 hundred cell phones are sold, the profit is given by P(23). B. When 23 hundred cell phones are sold, the revenue is given by P(23). C. When 23 hundred cell phones are sold, the cost is given by P(23).
Answers
(a) The profit function P(x) is obtained by subtracting the cost function C(x) from the revenue function P(x) = -0.05x^3 + 1.1x^2 + 201x - 200.
(b) To find the profit when x = 23 hundred cell phones are sold, substitute x = 23 into the profit function P(23) ≈ $748.95.
(c) Interpretation: P(23) represents the profit achieved when 23 hundred cell phones are sold.
(a) The profit function P(x) is calculated by subtracting the cost function C(x) from the revenue function R(x):
P(x) = R(x) - C(x) = (-1.9x^2 + 256x) - (0.05x^3 - 3x^2 + 55x + 200) = -0.05x^3 + 1.1x^2 + 201x - 200.
(b) To find the profit when x = 23 hundred cell phones are sold, substitute x = 23 into the profit function P(x):
P(23) = -0.05(23)^3 + 1.1(23)^2 + 201(23) - 200 ≈ $748.95.
(c) Interpretation: P(23) represents the profit achieved when 23 hundred cell phones are sold. The value obtained, $748.95, represents the difference between the revenue generated by selling 23 hundred cell phones and the cost incurred in producing and selling those phones. It indicates the financial gain or loss associated with selling that particular quantity of cell phones. In this case, a positive profit of $748.95 implies that selling 23 hundred cell phones resulted in a net financial gain for the business.
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You rent an apartment that costs $1200 per month during the first year, but the rent
is set to go up 12% per year. Write a recursive formula to show what the rent is during
the nth year of living in the apartment.
Answers
Answer:
So first we have to figure out what $1200 per month for 12 months is since it was a year. 1200x12=14,400 a year which is outrageous for an apartment just saying.
Then we find out what 12% of the 1200 is and then add it to 1200.
To find the percentage of that we need to know how many times 12 fits into 100% 12 fits into 100 a total of 8.33333333333 times.
So now we divide 1200 by 8.33333333333 to see what 12% of it is.
1200 divided by 8.33333333333 is 144. So 1200+144=1344.
Now since one of the nine years has passed already we only have 8 remaining.
Now we see what 1344 a month for 12 months is 16,128 a year.
Now we multiply 16,128 by 8 to get the outcome of 8 more years. So 16,128x8=129,024
Thats it pls mark brainliest
Step-by-step explanation:
Whoever helps me with this I will vemo or cashapp or paypal you $10 but if I get everything right ill paypal $20
Answers
Answer:
15. 2/6 or 1/3
18. 3/6 or 1/2
21. 25/26
29.
a- 1/12
50/600
b- 22/100
132/600
Step-by-step explanation:
The first ones are self explanatory lol.
to find the ones for 29, i just multiplied divided 12 by 600 and got 50 so that was my numerator and on the 2nd one i just multiplied 22 by 6 to get 132 because the denomenator is 600.
find the 10th and 75th percentiles for these 20 weights
29, 30, 49, 28, 50, 23, 40, 48, 22, 25, 47, 31, 33, 26, 44, 46,
34, 21, 42, 27
Answers
The 10th percentile is 22 and the 75th percentile is 46 for the given set of weights.
To find the 10th and 75th percentiles for the given set of weights, we first need to arrange the weights in ascending order:
21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 40, 42, 44, 46, 47, 48, 49, 50
Finding the 10th percentile:
The 10th percentile is the value below which 10% of the data falls. To calculate the 10th percentile, we multiply 10% (0.1) by the total number of data points, which is 20, and round up to the nearest whole number:
10th percentile = 0.1 * 20 = 2
The 10th percentile corresponds to the second value in the sorted list, which is 22.
Finding the 75th percentile:
The 75th percentile is the value below which 75% of the data falls. To calculate the 75th percentile, we multiply 75% (0.75) by the total number of data points, which is 20, and round up to the nearest whole number:
75th percentile = 0.75 * 20 = 15
The 75th percentile corresponds to the fifteenth value in the sorted list, which is 46.
Therefore, the 10th percentile is 22 and the 75th percentile is 46 for the given set of weights.
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what is the value of each region that makes the inequalities true 5- 2x < -3-3(-5+ x) > 3
Answers
Let us solve for x in each of the inequalities.
The first inequality.
\(5-2x<-3_{}\)subtracting 5 from both sides gives
\(5-2x-5<-3_{}-5\)\(-2x<-8\)finally, multiplying both sides by -2 reverse the direction of the inequality to give
\(-\frac{2x}{-2}>-\frac{8}{-2}\)\(\boxed{x>4}\)which is our answer!
The second inequality
\(-3\mleft(-5+x\mright)>3\)
Dividing both sides by -3 gives
\(\frac{3(-5+x)}{-3}<\frac{3}{-3}\)\(\rightarrow(-5+x)<-1\)finally, adding -5 to both sides gives
\(-5+x+5<-1+5\)\(\begin{gathered} \rightarrow x<-1+5 \\ \boxed{x<4} \end{gathered}\)which is our answer!
You decide to go to the gym with your mom and she teaches you how to workout with wall balls. This movement is where you squat and throw a weighted ball at a target on the wall 8-12 feet in the air, then catch it, squat and do it again. It works out multiple muscle groups.
Your mom gives you a couple of tips:
Tip #1: As with any weight, the farther you hold the ball, the heavier it will seem. So keep it close to your face and body. You don't want the ball to smash into your face, so concentrate on keeping your hands on the side and slightly toward the bottom of the ball.
Tip #2: Keep your hands high when you catch the ball. Its momentum will drive you down and your job is to use it to bounce back upwards again.
Tip #3: Stand the right distance from the wall. If you stand too far away, you will be doing a lot of extra work because you have to throw the ball forward and lose your upward momentum. Where you stand depends on how high you are throwing, generally about 15° when you are looking up at the target.
You ask your mom, "How do I know where to stand then?"
She said, "You figure it out. You are learning about trig ratios."
You decide to figure it out and sketch out the following drawing.
1) How far should you stand if you are hitting the target 8 feet high on the wall?
a) 0.03 ft.
b) 2.14 ft.
c) 2.07
d) 7.73
2) How far away should you stand if you are hitting the target 9 feet high on the wall?
a) 0.03 ft
b) 2.41 ft
c) 2.33 ft
d) 8.70 ft
3) How far away should you stand if you are hitting the target 10 feet high on the wall?
a) 2.59 ft
b) 9.66 ft
c) 2.68 ft
d) 0.03 ft
4) EXPLAIN your answers using one of the trigonometric ratios.
Answers
1. b) 2.14 ft.
2. b) 2.41 ft
3. c) 2.68 ft
4. Using the tangent ratio, tan ∅ = opposite side/adjacent side, each of the answers were determined.
How to Apply the Tangent Ratio?The tangent ratio is one of the trigonometric ratios that is used to solve a right triangle is we known the reference angle measure and the length of the adjacent or opposite sides.
The tangent ratio is given as, tan ∅ = opposite side/adjacent side.
To find how far away you stand if you are hitting the target at different heights on the wall, apply the tangent ratio.
1. ∅ = 15°
Opposite side = how far away you should stand
Adjacent side = 8 ft
Tan 15 = how far/8
How far you should stand = (8)(tan 15)
How far you should stand = 2.14 ft
2. ∅ = 15°
Opposite side = how far away you should stand
Adjacent side = 9 ft
Tan 15 = how far/9
How far you should stand = (9)(tan 15)
How far you should stand = 2.41 ft
3. ∅ = 15°
Opposite side = how far away you should stand
Adjacent side = 10 ft
Tan 15 = how far/10
How far you should stand = (10)(tan 15)
How far you should stand = 2.68 ft
4. Using the tangent ratio, tan ∅ = opposite side/adjacent side, each of the answers were determined.
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(02.01 LC)
What is the solution for the equation 6x - 8 = 4x? x = _. (Input whole number only).
Answers
Answer:
4
Step-by-step explanation:
start by subtracting 6x from 4x. that is -2x. then you have -8=-2x. divide the -2 out of the -2x from the -8. then you have 4=x
Some campers go out to collect
water from a stream. They share the water equally
among 8 campsites. How much water does each
campsite get? Bucket: 62.4 L
Answers
Each campsite will receive 7.8 liters of water.
If the campers collect water from a stream and share it equally among 8 campsites, we need to determine how much water each campsite receives.
The total amount of water collected is given as 62.4 liters in a bucket. To find the amount of water per campsite, we divide the total amount of water by the number of campsites.
Dividing 62.4 liters by 8 campsites gives us 7.8 liters per campsite.
It's important to note that this calculation assumes an equal distribution of water among all the campsites. However, in practical situations, the division may not be exact due to factors such as spillage, uneven pouring, or variations in the bucket's actual capacity.
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Help me please with this question:)
Answers
Answer:
it's the second one.
option 2