Answers
Using limits, it is found that the infinite sequence converges, as the limit does not go to infinity.
How do we verify if a sequence converges of diverges?Suppose an infinity sequence defined by:
\(\sum_{k = 0}^{\infty} f(k)\)
Then we have to calculate the following limit:
\(\lim_{k \rightarrow \infty} f(k)\)
If the limit goes to infinity, the sequence diverges, otherwise it converges.
In this problem, the function that defines the sequence is:
\(f(k) = \frac{k^3}{k^4 + 10}\)
Hence the limit is:
\(\lim_{k \rightarrow \infty} f(k) = \lim_{k \rightarrow \infty} \frac{k^3}{k^4 + 10} = \lim_{k \rightarrow \infty} \frac{k^3}{k^4} = \lim_{k \rightarrow \infty} \frac{1}{k} = \frac{1}{\infty} = 0\)
Hence, the infinite sequence converges, as the limit does not go to infinity.
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The series diverges by the comparison test.
We have for large enough \(k\),
\(\displaystyle \frac{k^3}{k^4+10} \approx \frac{k^3}{k^4} = \frac1k\)
so that
\(\displaystyle \sum_{k=0}^\infty \frac{k^3}{k^4+10} = \frac1{10} + \sum_{k=1}^\infty \frac{k^3}{k^4+10} \approx \frac1{10} + \sum_{k=1}^\infty \frac1k\)
and the latter sum is the divergent harmonic series.
Related Questions
Triangle ABC has vertices
A(-3, 3), B(2, 4), and C(-2,
2) and is translated
according to the rule:
(x, y) –> (x+2, y-4).
What are the coordinates
of the vertices of the
translated figure?
Answers
The coordinates of the translated triangle A'B'C' are: A'(-1, -1), B'(4, 0), and C'(0, -2).
To find the coordinates of the vertices of the translated figure, we simply apply the given translation rule to each vertex of the original triangle.
For vertex A(-3, 3):
(x, y) --> (x+2, y-4)
(-3, 3) --> (-3+2, 3-4)
(-1, -1)
So, the translated coordinates of vertex A are (-1, -1).
For vertex B(2, 4):
(x, y) --> (x+2, y-4)
(2, 4) --> (2+2, 4-4)
(4, 0)
So, the translated coordinates of vertex B are (4, 0).
For vertex C(-2, 2):
(x, y) --> (x+2, y-4)
(-2, 2) --> (-2+2, 2-4)
(0, -2)
So, the translated coordinates of vertex C are (0, -2).
Therefore, the vertices of the translated triangle are A'(-1, -1), B'(4, 0), and C'(0, -2).
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Which phrase represents the integer -13
Answers
Answer:
Thirteen degrees below zero
Step-by-step explanation:
Because it is the only measurement that can be described with a negative magnitude.
Derek borrows $30,924.00 to buy a car. He will make monthly payments for 6 years. The car loan has an interest rate of 5.89%. After a 11.00 months Derek decides to pay off his car loan. How much must he give the bank?
Answers
Derek must give the bank approximately $26,695.78 to pay off his car loan after 11 months. To calculate how much Derek must give the bank to pay off his car loan after 11 months, we need to consider the principal amount borrowed, the interest rate, the loan term, and the number of payments made.
Derek borrows $30,924.00 to buy a car, and he will make monthly payments for 6 years. The interest rate is 5.89%.
First, we need to calculate the monthly interest rate. We divide the annual interest rate by 12 (number of months in a year) and convert it to a decimal:
Monthly interest rate = 5.89% / 12 = 0.4908%
Next, we calculate the number of payments made up to the point when Derek decides to pay off the loan. In this case, he makes payments for 11 months.
Now, let's calculate the remaining balance on the car loan using the formula for the remaining balance on an amortizing loan:
Remaining balance = Principal * [(1 + r)^n - (1 + r)^p] / [(1 + r)^n - 1]
Where:
Principal = $30,924.00 (initial loan amount)
r = Monthly interest rate (0.004908)
n = Total number of payments (6 years * 12 months per year = 72 payments)
p = Payments made (11 payments already)
Substituting the values into the formula, we have:
Remaining balance = $30,924.00 * [(1 + 0.004908)^72 - (1 + 0.004908)^11] / [(1 + 0.004908)^72 - 1]
Using a calculator, we can evaluate this expression:
Remaining balance ≈ $30,924.00 * [1.4042 - 1.0552] / [1.4042 - 1] ≈ $30,924.00 * 0.3490 / 0.4042 ≈ $26,695.78
Therefore, Derek must give the bank approximately $26,695.78 to pay off his car loan after 11 months.
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what is the answer 6u+2(u-8) - 4
Answers
Answer:
8u - 20
Step-by-step explanation:
on exploration 4.3.3, what is a vertical asymptote for question 2?
Answers
A vertical asymptote is a line on the x-axis where the graph of a function approaches infinitely close but does not actually reach. In this case, the vertical asymptote is x = 4.
In exploration 4.3.3, the vertical asymptote for question 2 is at x = -2. Let's see what is meant by a vertical asymptote.What is a vertical asymptote?A vertical asymptote is a line that a function approaches but never touches.
The curve gets arbitrarily close to the vertical line but never touches it. When the function approaches a vertical asymptote, the function approaches infinity or negative infinity.For instance, consider the function f(x) = 1/x. It has a vertical asymptote at x = 0. This is because the function's value becomes increasingly larger as x approaches 0 from the positive side, and the value of the function becomes increasingly smaller as x approaches 0 from the negative side.Therefore, if a curve goes on increasing or decreasing on both sides of a vertical line (called a vertical asymptote), the curve approaches that line but never touches it. This is the basic idea of a vertical asymptote.What is the vertical asymptote for question 2?The rational function f(x) = (x + 2) / [(x + 2)^2 - 4] is being studied in question 2 of exploration 4.3.3. It can be seen that (x + 2)^2 - 4 = (x + 2 - 2)(x + 2 + 2) = (x)(x + 4).Therefore, the denominator (x + 2)^2 - 4 can be written as (x)(x + 4).The vertical asymptotes of a rational function occur where the denominator equals zero. Since the denominator is (x)(x + 4), the vertical asymptotes occur where x = 0 or x = -4.However, x = 0 is not a vertical asymptote because there is a hole in the graph. So, the only vertical asymptote is at x = -2. This means that the curve approaches x = -2 but never touches it. Therefore, the vertical asymptote for question 2 in exploration 4.3.3 is x = -2.
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The asymmetric cryptography algorithm most commonly used is:
O GPG
O RSA
O ECC
O AES
Answers
Answer
Step-by-step explanation:
What is the equation of a vertical line through the point (3, 2)?
Answers
Answer:
If you consider what the slope looks like for a vertical line, it is an infinite slope. What that means is that you cannot have a y-value in your equation:
x = 3
This x-value would be a vertical line that passes through all y-values for x = 3. So you can replace them in (3, y) with any value, and it will be included in the linear equation x = 3.
All 60 students in the seventh grade must choose a fall sport in which they will participate. If 2/5 of them choose volleyball, 25% of them choose soccer, and the remainder choose softball, how many students choose softball?
Answers
Answer:
21
Step-by-step explanation:
24 play volleyball
15 play soccer
Softball= 60-(24+15)=21
or
2/5 = 40%
100% -(40+25)= 35%
60x0.35=21
Al repartir pastelitos entre 7 niños cada uno le tocaron un pastelito completo y 3/7 de otro cuanto pastelitos reparto?
Answers
Answer: 10 magdalenas
espero que esto ayude
A box holds 120 apples. 8 families bought 2 boxes of apples to be shared equally among them. Each family has 5 members. how many apples did each family member recieve?
Answers
A box holds 120 apples where 8 families bought 2 boxes of apples. therefore, each family member recieve 55 apples.
What is the unitary method?The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value. If an event can occur in m different ways and if following it, a second event can occur in n different ways, then the two events in succession can occur in m × n different ways.
We have been given that box holds 120 apples where 8 families bought 2 boxes of apples to be shared equally among them. and Each family has 5 members.
Total number of member 8 families have = 8 x5 = 40
2 boxes of apples = 120 x 2 = 440 apples
therefore, each family member recieve the number of apples
440 / 8 = 55 apples
Hence, box holds 120 apples where 8 families bought 2 boxes of apples. therefore, each family member recieve 55 apples.
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A basketball team has 15 players on its roster. How many possible starting lineups does the team have? (There are 5 players on a basketball court.)
120
1,200
3,003
5,576
360,360
Answers
Answer:
3003
Step-by-step explanation:
This is just basically choosing 5 players out of 15 people.
15 choose 5 is \(\frac{15!}{10!*5!}\)
\(\frac{15*14*13*12*11}{5*4*3*2}\)
3*7*13*11=1001*3=3003
Feel free to tell me if I did anything wrong! :)
The basketball team has 3,003,600 possible starting lineups. Given that there are 15 players on the roster and 5 players on the basketball court, this calculation involves determining the number of ways to select 5 players out of the total 15.
To calculate the number of possible starting lineups, we can use the concept of combinations. We have 15 players on the roster, and we need to select 5 players to form a starting lineup.
The number of ways to choose 5 players out of 15 can be calculated using the combination formula:
C(n, r) = n! / (r! * (n - r)!)
In this case, n represents the total number of players (15) and r represents the number of players needed for the lineup (5).
Plugging these values into the formula, we get:
C(15, 5) = 15! / (5! * (15 - 5)!)
= (15 * 14 * 13 * 12 * 11 * 10!) / (5! * 10!)
= (15 * 14 * 13 * 12 * 11) / (5 * 4 * 3 * 2 * 1)
= 3,003,600
Therefore, there are 3,003,600 possible starting lineups for the basketball team. This means that the team has a vast number of combinations to choose from when selecting the starting lineup from the 15 players on its roster.
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find f(s). ℒ{(t − 1)?(t − 1)}
Answers
The answer for the given Laplace Transform, f(s), is e^(-s)[(1/s^2) - (1/s)].
To find f(s), we can start by applying the Laplace transform to the given function:
ℒ{(t − 1)(t − 1)} = ℒ{t^2 - 2t + 1} = 1/s^3 - 2/s^2 + 1/s
Identify the function
We have the function (t - 1)u(t - 1), where u(t - 1) is the unit step function.
Apply the Laplace Transform property for the unit step function
The Laplace Transform of u(t - a)f(t - a) is given by e^(-as)F(s), where a = 1 in our case. So, we need to find the Laplace Transform of f(t - a) = f(t - 1).
Find the Laplace Transform of f(t - 1)
Our function f(t - 1) = t - 1. The Laplace Transform of t^n is given by n!/(s^(n+1)), so for n = 1, the Laplace Transform of t is 1!/(s^2) = 1/s^2. Since we have t - 1, we find the Laplace Transform of the constant 1 as well, which is 1/s.
So, the Laplace Transform of f(t - 1) = (1/s^2) - (1/s).
Apply the Laplace Transform property
Now, we apply the e^(-as)F(s) property with a = 1:
f(s) = e^(-s)[(1/s^2) - (1/s)]
So, the answer for the given Laplace Transform, f(s), is e^(-s)[(1/s^2) - (1/s)].
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Round 4,482,186 to the hundreds place
Answers
Answer:
4,482,186 becomes 4,482,190
Step-by-step explanation:
Determine the place value of the digit 3 in the whole number. 30,500,421
Answers
Answer:
30,000,000 or Thirty million -------> Ten millions place
Step-by-step explanation:
Hope this helps!
Let f be a differentiable function such that f (2) = 4 and f (2) = − 1/2 . What is the approximation for f (2.1) found by using the line tangent to the graph of f at x = 2 ?
Answers
Using line tangent, the approximation for f(2.1) is 3.95
Given,
The point (a, f(a)) is on the line tangent to the graph of y = f(x) at x = a, which has a slope of f'(a).
The equation be like;
y - f(a) / (x - a) = f'(a)
y = f'(a) (x - a) + f(a)
Using the provided data and a = 2, we can determine that the tangent line to the graph of y = f(x) at x = 2 has equation
y = f'(2) (x - 2) + f(2)
y = -1/2 (x - 2) + 4
To compute a "approximation of f(2.1) using the line tangent to the graph of f at x = 2," one must substitute x = 2.1 for f in the equation for the tangent line (2.1). You get 2.1 when you plug this in.
y = -1/2 (x - 2) + 4
y = -1/2 (2.1 - 2) + 4
y = -1/2 x 0.1 + 4
y = 3.95
That is,
The approximation for f(2.1) using line tangent is 3.95
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The Product of 4 and z is the same as 16
Answers
Answer:
The variable can be any letter.
Step-by-step explanation:
The value of z is 4.
Given that,
The Product of 4 and z is the same as 16Based on the above information, the calculation is as follows:
4z = 16
z = 4
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Travis needs to mail 12 letters and 25 postcards. A letter needs a 79-cent stamp and a postcard needs a 30-cent stamp. How much money will Travis spend for postage?
Answers
Answer:
$16.98
Step-by-step explanation:
First you have to multiply the amount of letters and postcards by the amount each stamps costs:
12x0.79=9.48
25x0.30=7.50
Then add those values together to get the total:
9.48+7.50=16.98
$16.98 is your answer
Hope this helps!
Please answer correctly !!!!! Will mark brainliest !!!!!!!!!!!
Answers
Answer:
x5
Step-by-step explanation:
Answer:
(x - 1) (x + 4)
Step-by-step explanation:
D(-3,4) E(7, 6) 1:3
Answers
Answer:d
Step-by-step explanation:
Liz makes and sells necklaces. It costs her $40 for the supplie s to make 50 necklaces. How much must she sell each one for in order to make a profit of at least $250?
Answers
Using proportions, it is found that she must sell each necklace for at least $5.80 in order to make a profit of at least $250.
What is a proportion?A proportion is a fraction of a total amount
In this problem, it costs her $40 to make 50 necklaces. She wants a profit of at least $250, hence the 50 necklaces should sell for at least $250 + $40 = $290.
Hence, the minimum price for each one is given by:
$290/50 = $5.80.
She must sell each necklace for at least $5.80 in order to make a profit of at least $250.
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f f(x) = 16x – 30 and g(x) = 14x – 6, for which value of x does (f – g)(x) = 0? –18 –12 12 18
Answers
The value of 'x' that makes (f - g)(x) equal to zero, the answer is x = 12.
To find the value of 'x' for which (f - g)(x) = 0, we need to determine the value of 'x' that makes the difference between f(x) and g(x) equal to zero.
Given:
f(x) = 16x - 30
g(x) = 14x - 6
To calculate (f - g)(x), we subtract g(x) from f(x):
(f - g)(x) = f(x) - g(x)
= (16x - 30) - (14x - 6)
= 16x - 30 - 14x + 6
= 2x - 24
We set (f - g)(x) equal to zero and solve for 'x':
2x - 24 = 0
Adding 24 to both sides of the equation:
2x = 24
Dividing both sides by 2:
x = 12
The solution is x = 12 for the value of "x" that causes (f - g)(x) to equal zero.
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The table below shows the linear relationship between the number of people at a picnic and the total cost of the picnic.
Answers
Someone PLEASE PLEASEE help me asap i dont have time. I need the answers today please im really struggling right now. 50 points to answer. Please no wrong answers or guesses.
(-2, 3)
(2, -3)
(-3, -2)
(3, -2)
Answers
it definitely should be (-2,3)
2 + 3 + 1 17/20 + 1/2 + 3/4
Answers
Answer:
Step-by-step explanation:
12.1
Answer:
8 \(\frac{1}{10}\)
Step-by-step explanation:
\(\frac{17}{20}\) = \(\frac{34}{40}\) Multiplied the numerator and denominator by 2
\(\frac{1}{2}\) = \(\frac{20}{40}\) Multiplied the numerator and denominator by 20
\(\frac{3}{4}\) = \(\frac{30}{40}\) Multiplied the numerator and denominator by 10
2 + 3 + 1 \(\frac{34}{40}\) + \(\frac{20}{40}\) + \(\frac{30}{40}\)
6 \(\frac{84}{40}\) This can be written
6 + \(\frac{40}{40}\) + \(\frac{40}{40}\) + \(\frac{4}{40}\) another name for \(\frac{40}{40}\) is 1
6 + 1 + 1 +\(\frac{4}{40}\)
8\(\frac{4}{40}\) Divide the numerator and denominator by 4 to simplify
8 \(\frac{1}{10}\)
PLEASE HELP AND SHOW WORK ILL GIVE BRAINLIEST
Answers
Answer:
The number 7 is a rational number. Rational numbers are defined as numbers that result when two integers are divided.
Step-by-step explanation:
Lily needed to get her computer fixed. She took it to the repair store. The technician at the store worked on the computer for 3.5 hours and charged her $159 for parts. The total was $456.50. Write and solve an equation which can be used to determine x x, the cost of the labor per hour.
Answers
456.50=159 + 3.5x
OR
3.5x=456.50-159
Reason
456.50 is the total
159 is for the parts
3.5 is the number of hours
So to find the cost you have to find out how much it cost for the labor by subtracting the total cost from the cost of the parts. To find the cost of labor for each hour you have to divide the total cost of labor by the number of hours used to work on the computer
The equation to represent the problem is written as
$456.50 = 3.5x + $159
the cost of the labor per hour is $85.
How to determine the cost of the labor per hourLet x be the cost of the labor per hour.
The technician worked on the computer for 3.5 hours, so the labor cost would be 3.5x.
The total cost, including the labor and parts, was $456.50.
So, we can set up an equation to represent the total cost:
Total cost = Labor cost + Parts cost
$456.50 = 3.5x + $159
Solving for x:
Subtract $159 from both sides:
$456.50 - $159 = 3.5x
$297.50 = 3.5x
Finally, divide both sides by 3.5 to find the value of x:
x = $297.50 / 3.5
x = $85
Therefore, the cost of the labor per hour is $85.
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David made an error when determining the product of (–2.5)(–8.3).
Step 1: (–2.5)(–8.3) = (–8.3)(–2.5)
Step 2: = (–8.3)(–2) + (–8.3)(–0.5)
Step 3: = (–16.6) + (–4.15)
Step 4: = –20.75
In which step did David make his first error?
Answers
Answer:
step 3
Step-by-step explanation:
when you multiply an even amount of negative numbers you will always get a positive result thus the answe would be 16.6 + 4.15 as in positive numbers
nationally, the proportion of red cars on the road is 0.12. a statistically-minded fan of the philadelphia phillies wonders if phillies fans are more likely to drive red cars. one day during a home game, he takes an srs of 210 and counts 35 red cars. what is the test statistic?
Answers
The test statistic for this scenario is 1.14. To calculate the test statistic, we can use the formula:
(test statistic) = (sample proportion - hypothesized proportion) / standard error
Here, the hypothesized proportion is the national proportion of red cars on the road, which is 0.12. The sample proportion is the proportion of red cars in the sample, which is 35/210 = 0.167.
To calculate the standard error, we use the formula:
\(standard error=\sqrt{\frac{(hypothesized proportion * (1 - hypothesized proportion)) }{ sample size} }\)
Plugging in the values, we get:
standard error = √[(0.12 * (1 - 0.12)) / 210] = 0.032
Now, we can calculate the test statistic:
test statistic = (0.167 - 0.12) / 0.032 = 1.14
The test statistic tells us how many standard errors the sample proportion is away from the hypothesized proportion. In this case, the test statistic is 1.14, which means that the sample proportion of Phillies fans driving red cars is 1.14 standard errors away from the national proportion of red cars on the road. We would need to compare this test statistic to a critical value from a t-distribution to determine if the difference is statistically significant at a given level of significance.
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help help guys show your solution and also answer the question
Answers
All answers are in the picture below!
Which number is the decimal form of 3/8
Answers
Answer: 0.375
Step-by-step explanation: