In the factorization, the bottom triangular matrix L is the only one.
What is Cholesky decomposition?The exponent of a number indicates how many times it has been multiplied by itself.
Exponent-related issues can be solved using either exponent laws or exponent characteristics. When a number is repeated numerous times by itself, writing the product without the use of exponents becomes very difficult. Major exponentiation rules are also thought of as having these characteristics.
Given that A is a positively defined symmetric matrix with a Cholesky decomposition,
A = L[tex]L^{T}[/tex] = [tex]R^{T} R[/tex]
Let A be our positive define symmetric matrix,
suppose A has Cholesky decomposition,
A = L₁L₂ = L₁L₂ , for L₁, L₂ lower triangular matrix with positive diagonal entries,
thus (A x ,x) = (Lx, Lx) = (Lx, Lx)
pick x = eₓ the last coordinate vector,
then (Aₓ, x) = Axx = || L₁x||² = || L₂x||²
Given that L1, and L2 are lower triangular, which necessitates that their lower right entry be the same,
The Kth entry is currently the last row.
given by (L₁, eₓ, eₓ) = 1/√Axx (L₁eₓ, L₂eₓ) = 1/ Aₓ = (Aeₓ, eₓ)
so the last row of L₁, L₂ be the same,
we reduce our A to new (n - 1)x(n - 1) submatrix,
which will likewise be true, and then repeat the process to discover each row's unique value.
Thus, the inductive process shows that L1 = L2.
As a result, the factorization's bottom triangular matrix L is exclusive.
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Colette bought 1/4 of a pound of green grapes and 1/8 of a pound of red grapes. How much more did the green grapes weigh than the red grapes?
The green grapes weigh 0.125 pounds more than the red grapes.
Step-by-step explanation:1/4 of a pound = 0.25 pounds
1/8 of a pound = 0.125 pounds
0.25 - 0.125 = 0.125 pounds
Hope this helps :)
Question 1 of 9
Use the scale to help you solve the equation and find the value of x. Enter the
value of x below. x+8=10
Answer: x + 8 - 8 = 10 - 8
Simplifying the left side of the equation gives us:
x = 2
Therefore, the value of x in this equation is 2.
Suppose the COMBINED area is known to be 0.10416, assume there is equal area on each side. Determine the corresponding Z-scores.A) z=±1.62 B) z=±1.58 C) z=±1.72 D) z=±1.66 E) z=±1.69 F) z=±1.49 G) None of These
An combined area of 0.10416 to the right means that it must have an area of 0.90 to the left, so the answer is again 1.28.
Every normally distributed random variable has a slightly different distribution shape, the only way to find areas using a table is to standardize the variable - transform our variable so it has a mean of 0 and a standard deviation of 1. How do we do that? Use the z-score!
Z = ( x - μ)/σ
if the random variable X has a mean μ and standard deviation σ,
then transforming X using the z-score creates a random variable with mean 0 and standard deviation 1!
With that in mind, we just need to learn how to find areas under the standard normal curve, which can then be applied to any normally distributed random variable.
An area of 0.10416 to the right means that it must have an area of 0.90 to the left, so the answer is again 1.28.
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Every normally distributed random variable has a slightly different distribution shape, the only way to find areas using a table is to standardize the variable - transform our variable so it has a mean of 0 and a standard deviation of 1. How do we do that? Use the z-score!
Z = ( x - μ)/σ
if the random variable X has a mean μ and standard deviation σ,
An area of 0.10416 to the right means that it must have an area of 0.90 to the left, so the answer is again 1.28.
Are y = 6 and x = - 1 perpendicular lines?
answer: yes.
since y = 6 and x = -1 are parallel to the y and x axis, respectively, they hit each other at a 90° angle when graphed, making a plus sign. they are perpendicular lines.
On January 1, 2019, Marblehead, Massachusetts had a population of approximately 19,800 people. On that date, 1,098 Marblehead residents were living with asthma. Between January 1, 2019 and September 1, 2019, there were 495 newly diagnosed cases of asthma in Marblehead. You can assume that the population (19,800) remained constant throughout 2019.
What was the prevalence of asthma in Marblehead on January 1, 2019?
What was the prevalence of asthma in Marblehead on September 1, 2019?
What was the incidence of asthma in Marblehead between January 1, 2019 and September 1, 2019?
Based on the information provided above, it is NOT possible to calculate asthma incidence rate in Marblehead for the year 2018.
The prevalence of asthma on January 1, 2019 is 0.055, the prevalence of asthma on September 1, 2019 is 0.080, the incidence of asthma between January 1 and September 1 is 0.025, there is not possible to calculate asthma incidence rate for the year 2018.
How to calculate prevalence and incidence asthma?Prevalence is the formula to determine the rate of some case based on population. This can be calculated by divide the case by the population.
Incidence rate is same as prevalence in some factor except it only calculate time range and for this case it have same formula which is we can calculate incidence rate by divide the case by the population.
So, for January 1, 2019 is
Prevalence = 1,098 / 19,800
= 0.055
For September 1, 2019 is
Prevalence = (1,098 + new 495) / 19,800
= 1,593 / 19,800
= 0.080
For incidence of asthma between January 1 and September 1 is
Incidence = 495 / 19,800
= 0.025
For 2018 because we are not provided with data for that year so it is impossible to calculate it.
Thus, the prevalence in January 1 is 0.055, September 1 is 0.080, and incidence between January 1 and September 1 is 0.025, and also there is not possible to calculate asthma incidence rat for the year 2018.
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A cruise ship traveled to Madagascar and
back. On the trip there it traveled 12 km/h
and on the return trip it went 18 km/h.
How long did the trip there take if the
return trip took six hours?
A) 12 hours
C) 9 hours
B) 10 hours
D) 7 hours
To find the time it took the cruise ship to travel to Madagascar, we need to use the formula for distance, which is distance = speed x time. We are given the distance and the speed of the ship on the trip there, so we can solve for the time by dividing the distance by the speed:
time = distance / speed
We are told that the distance of the trip there was the same as the distance of the return trip, which took 6 hours. We are also told that the ship traveled 12 km/h on the trip there, so the distance of the trip there was 12 km/h x 6 h = 72 km.
Now we can plug this information into the formula to solve for the time:
time = 72 km / 12 km/h = <<72/12=6>>6 hours
Therefore, it took the cruise ship 6 hours to travel to Madagascar. The correct answer is therefore (A) 12 hours.
5k^2+20k-25=0
please i need help
The value of k is 1 and -5.
What is factorization?
A number or other mathematical object is factorized or factored when it is written as the product of numerous factors, typically smaller or simpler things of the same kind.
Here, we have
Given function: 5k²+20k-25 = 0
Now, we factorize the given function and we get
5k²+20k-25 = 0
5(k²+4k-5) = 0
k² + 4k - 5 = 0
k² + 5k - k -5 = 0
k(k+5) -1(k+5) = 0
(k+5)(k-1) = 0
k+5 = 0, k-1 = 0
k = -5, k = 1
Hence, The value of k is 1 and -5.
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Find the greatest possible error :19oz
Answer:
0.5 oz
Step-by-step explanation:
You want the greatest possible error associated with the measurement 19 oz.
Measurement errorThe possible error is generally considered to be half the place value of least-significant digit. Here, the least significant digit has a place value of units, so the greatest error is 1/2 unit, or 0.5 ounces.
__
Additional comment
We know that values of 18.5 oz up to 19.4999... oz will round to 19 oz. That is why the greatest error is considered to be 0.5 oz.
In practice, there can be some difficulty associated with knowing that a value is 18.5 instead of 18.49. Hence real-world error may actually be slightly larger than 0.5 oz. In math, we ignore that case.
Harry and Tim both made New Year’s Resolution. Harry made 5 more resolutions than Tim. Together they made 13 resolutions. How many resolutions did Harry make?
Answer:
Harry made 9 resolutions.
Step-by-step explanation:
Tim made x resolutions.
Harry made five more, x + 5.
Harry and Tim's together was 13.
x + x + 5 = 13
combine like terms.
2x + 5 = 13
subtract 5
2x = 8
divide by 2
x = 4
Tim made 4 resolutions.
Harry made 4 + 5, that is, 9 resolutions.
Check:
Harry and Tim together is 13.
4 + 9 = 13
Harry made 9 resolutions.
What is the sum of the interior angles of a regular polygon with 9 sides?
1,260°
1,620°
1,800°
1,980°
The sum of the interior angles of a regular polygon with 9 sides will be 1260°. The correct option is A.
What is the angle of the polygons?An angle is defined in mathematics as the figure formed by joining two rays at their common endpoint. An interior angle is an angle that exists within a shape. Polygons are closed shapes with sides and vertices. All of the interior angles of a regular polygon are equal.
Given that the number of the sides of the polygon is 9. The sum of the interior angles of the polygons will be calculated as,
Sum = ( n - 2 ) 180°
Sum = ( 9 - 2 ) 180
Sum = 7 x 180
Sum = 1260°
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The terminal side of an angle θ in standard position intersects the unit circle at (√30/7,-√19/7). What is cot(θ)?
The cotangent of the angle is -√570/30
How to determine the cotangent of the angle?From the question, we have the following parameters that can be used in our computation:
(√30/7,-√19/7)
This means that
(x, y) = (√30/7,-√19/7)
The cot(θ) is calculated as
cot(θ) = y/x
Substitute the known values in the above equation, so, we have the following representation
cot(θ) = (-√19/7)/(√30/7)
Evaluate
cot(θ) = -√19/√30
Rationalize
cot(θ) = -√570/30
Hence, the value of cot(θ) is -√570/30
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What is the y/x ratio
The angles whose measure are greater than the measure of ∠1 are ∠2, ∠3, ∠7, ∠5 .
What is the general equation of a straight line?The general equation of a straight line is -
y = mx + cwhere -
[m] : slope[c] : y - interceptGiven is a graph as shown in the image.
The ratio of {y/x} is called the slope of a line. So, we can write -
m = 3/2
m = 1.5
Therefore, the ratio of [y] by [x] is equivalent to 1.5.
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Which of the following expressions has the leftmost value on the number line?
A.-7-10 O
B. 7-10
C.10-(-7)
D.10-7
Answer:
A.-7-100. [Since it is -107]
karla is offered a job as parking lot attendant. the job will pay 2400 per month, paid biweekly. along with the base salary, the company offers to pay half the cost of medical insurance and will match karlas contribution to a retirment plan up to a total of 1500. if the full cost of the medical insurance is 450 a month and karla plans to contribute the full 1500 every year tpa retirement plan, what is the actual annual value of this job to karla?
The actual annual value of this job to Karla is 33000.
What are arithmetic operations?
Arithmetic operations is a branch of mathematics that studies numbers and the operations on numbers that are useful in all other branches of mathematics. It consists primarily of operations like addition, subtraction, multiplication, and division.
We have,
the job will pay 2400 per month,
if the full cost of the medical insurance is 450 a month and Karla plans to contribute the full 1500 every year tpa retirement plan,
the actual annual value of this job to Karla is:
2400 per month
annual value = 2400 * 12 = 28,800
the medical insurance is 450 a month
half the cost of medical insurance is 225 a month
annual value = 225 * 12 = 2700
the full 1500 every year tpa retirement plan,
So, the total value is:
= 28,800 + 2700 + 1500
= 33000
Hence, the actual annual value of this job to Karla is 33000.
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Keyshawn is asked to compare and contrast the domain and range of two functions F(x)=5x g(x)=5^x which statements could he include in his explanation
Domain is the input and range is the output of a function. By considering the options The domain of both function is all real number and The range of g(x) >0 are the correct answers.
What is Domain?Domain is the input of a function.
What is range?Range is the output of a function.
The domain of both functions is the set of all real numbers, since both functions are defined for all values of x.
The range of the function g(x)=5^x is the set of all positive real numbers, since any positive real number can be obtained by raising 5 to the power of a real number
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There were 616 tickets purchased for a major league baseball game. The lower box tickets cost $12.50 and the upper reserved tickets cost $8.00. The total amount of money spent was $6543.50. How many of each kind of ticket were purchased?
The number of the tickets purchased for upper reserve is 146 and lower is 470.
What is an expression?
The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that there were 616 tickets purchased for a major league baseball game. The lower box tickets cost $12.50 and the upper reserved tickets cost $8.00. The total amount of money spent was $6543.50.
The equations will be formed as,
L + U = 616
12.5L + 8U = 6543.5
Solve the above equations,
12.50L + 8 (616-L) = 6543.5
12.5L + 4928 - 8L = 6543.5
12.50L - 8L = 6543.5 - 4428
4.50L = 2118.5
L = 470
U = 616 - 470
U = 146
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assume that the maximum speed your car will go is a linear function of the steepness of the hill is going up or down. SUppose the car can go a maximum of 55 mph up a 5 hill and a maximum of 104 down a 2 hill
The maximum speed of the linear function is y = -7x + 90
Assume that the maximum speed your car will go is a linear function of the steepness of the hill is going up or down?a. we have two points connecting (speed, degree) are (+5,55) and (-2,104) representing (x₁ , y₁) and (x₂, y₂)
Slope = (y₂ - y₁) / (x₂ - x₁)
slope = (104-55) / (-2-5) = 49/-7 = -7 mph/1°
speed = slope x (degree) + b
Solve 'b':
55 = -7 x (5) + b
55 = -35 + b
b = 55 + 35
b = 90
y₁ = -7x₁ + 90
b) What does the speed-intercept equal and what does it represent?
The speed is 90 mph when there is no hill.
c ) What does the degree of steepness-intercept and what does it represent?
Up the hill, the steepness is roughly 12.9 degrees at its maximum. Whenever there is a slope, the car will not move.
d) Interpret the slope in words.
The speed drops by 7 mph for every 1 degree that the hill's steepness increases.
e) If your top speed is 83 mph, how steep is the hill? Be sure to include whether the direction is up or down.
83 = -7 x + 90
83 -90 = -7x
-7 = -7x
x = 1° up the hill
f) How fast could you go down a 7º hill?
y = -7 (-7) + 90
y = 139 mph
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Leila works mowing lawns and babysitting.She earns $8.80 an hour for mowing and $7.30 and hour and babysitting.How much will she earn for 1 hour of mowing and 5 hours of babysitting?
Answer: $45.3
Step-by-step explanation:
1 hour of mowing: $8.80
5 hours of babysitting: $7.30 x 5 = $36.5
$36.5 + 8.80 = $45.3
$45.3 is your answer.
Answer:
45.30
Step-by-step explanation:
its simple math bro lol
A train travelling along a straight line with constant acceleration is observed to travel consecutive distances of 1km in times of 30s and 60s respectively. Find the initial velocity of the train.
Answer:
116.7m/s
Step-by-step explanation:
distance(s) = 1km = 1 × 1000m = 1000m
Time(t) = 60– 30 = 30s
initial velocity = u
acceleration (a) = 10m/s
using the formula
S = ut + ½at²
1000 = 30u + ½ × 10 × 30²
1000 = 30u + 5 × 900
1000 = 30u + 4500
30u = 4500 – 1000
30u = 3500
divide both sides by 30
u = 116.7m/s
i hope i helped
What is the equation in standard form of the line that passes through the
point (4, −8) and 1 has a slope of 1/4?
The equation in standard form of the line that has a slope of 1/4 and passes through the point (4, −8) is written as: -x + 4y = -36.
What is the Equation of a Line in Standard Form?The equation of a line can be written in standard form as Ax + By = C.
Given the following:
A point (4, -8) = (a, b).
Slope (m) = 1/4.
Substitute m = 1/4 and (a, b) = (4, -8) into y - b = m(x - a):
y - (-8) = 1/4(x - 4)
y + 8 = 1/4(x - 4)
Rewrite the equation in standard form:
4(y + 8) = x - 4
4y + 32 = x - 4
4y = x - 4 - 32
4y = x - 36
-x + 4y = -36
The equation is: -x + 4y = -36.
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please help with this
The equation variation for the electric circuit is V = 25R.
How to find the equation variation of an electric circuit?In each electric circuit, the voltage in a battery is measured in volt and the resistance in the circuit is measured in ohms.
The voltage, V is directly proportional to the resistance, R. In a given circuit the voltage is 200 volts, when the resistance is 8 ohms.
The equation variation for the circuit can be represented as follows:
V α R
V = kR
where
k = constant of proportionalityTherefore,
voltage = 200 volts
resistance = 8 ohms
200 = 8k
k = 200 / 8
k = 25
Therefore, the equation is V = 25R
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Determine which of the lines, if any, are parallel or perpendicular. Explain.
6. Line a passes through (0, 4) and (4, 3).
Line b passes through (0, 1) and (4, 0).
Line c passes through (2, 0) and (4, 4).
Answer:
Step-by-step explanation:
c 0,4 .01
Can someone please answer this problem for me.
Answer:
$1 per movie
$5.50 per video game
Step-by-step explanation:
We know 5 movies and 2 video games cost $16 and 3 movies and 8 video games cost $47.
Let x be the cost of 1 movie and y be the cost of a video game.
5x + 2y = 16
3x + 8y = 47
In the first equation, lets solve for y
2y = 16 - 5x
y = 8 - 2.5x
Now, we can substitute this into the other equation.
3x + 8(8 - 2.5x) = 47
3x + 64 - 20x = 47
-17 x = - 17
x = 1
So a movie costs $1 to rent.
2y = 16 - 5(1)
2y = 11
y = 5.5
A video game costs $5.50 to rent.
Create a multiplication equation with a solution of 2/7
Answer:
1 / 7 · 2 = 2 / 7
What does MW equal or mean
The value of MW in the given figure is 5.
What is meant by an angle?In Euclidean geometry, an angle is a figure made up of two rays that have a shared vertex, also known as a common terminus, and are referred to as the angle's sides. The angles of two rays are situated in the plane containing the rays. An angle is also created when two planes intersect at an angle. These are referred to as dihedral angles. Another technique to define two curves is to specify the angle generated by rays that are tangent to the intersection of two curves. Angle is another word for how long a rotation or angle is.
Given,
In the figure,
PM=MW
It means PM is similar to MW
As given in the figure,
PM=5
Since PM is similar to MW
Then,
MW=5
Therefore, the value of MW in the given figure is 5.
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NO LINKS!!
Rewrite the logarithm as a ratio of common logarithms and natural logarithms.
log_1/3(4)
a. common logarithms
b. natural logarithms
Common logarithm is the logarithm with the base 10, it is written as:
log a, lg a or log₁₀ aNatural logarithm is the logarithm with the base of e and is written as ln a.
How to rewrite a logarithm [tex]log_a\ b[/tex] as a ratio of logarithms with a different base c:
[tex]log_a\ b=log_c\ b\ /\ log_c\ a[/tex]Apply this to the given logarithm:
a. Common logarithms:
[tex]log_{1/3}\ 4=log\ 4\ /\ log\ (1/3)[/tex]b. Natural logarithms:
[tex]log_{1/3}\ 4=ln\ 4\ /\ ln\ (1/3)[/tex]Answer:
[tex]\textsf{a)} \quad \dfrac{\log_{10}4}{\log_{10}\dfrac{1}{3}}=-\dfrac{\log_{10}4}{\log_{10}3}[/tex]
[tex]\textsf{b)} \quad \dfrac{\ln 4}{\ln \dfrac{1}{3}}=-\dfrac{\ln 4}{\ln 3}[/tex]
Step-by-step explanation:
Given logarithm:
[tex]\log_{\frac{1}{3}}(4)[/tex]
Part (a)The common logarithm is the logarithm with base 10.
[tex]\boxed{\textsf{Change of base}: \quad \log_ba=\dfrac{\log_xa}{\log_xb}}[/tex]
Use the change of base formula to rewrite the given logarithm as a ratio of common logarithms:
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\log_{10}4}{\log_{10}\dfrac{1}{3}}[/tex]
[tex]\boxed{\begin{minipage}{5cm} \underline{Log quotient law}\\\\$\log_a \left(\dfrac{x}{y}\right)=\log_ax - \log_ay$\\\end{minipage}}[/tex]
This can be simplified using the log quotient rule:
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\log_{10}4}{\log_{10}\dfrac{1}{3}}[/tex]
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\log_{10}4}{\log_{10}1-\log_{10}3}[/tex]
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\log_{10}4}{0-\log_{10}3}[/tex]
[tex]\implies \log_{\frac{1}{3}}(4)=-\dfrac{\log_{10}4}{\log_{10}3}[/tex]
Part (b)The natural logarithm is the logarithm with base e.
[tex]\textsf{Also} \; \log_ex=\ln x[/tex]
Use the change of base formula to rewrite the given logarithm as a ratio of natural logarithms:
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\log_{e}4}{\log_{e}\dfrac{1}{3}}[/tex]
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\ln 4}{\ln \dfrac{1}{3}}[/tex]
[tex]\boxed{\begin{minipage}{5cm} \underline{Log quotient law}\\\\$\ln \left(\dfrac{x}{y}\right)=\ln x - \ln y$\\\end{minipage}}[/tex]
This can be simplified using the log quotient rule:
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\ln 4}{\ln \dfrac{1}{3}}[/tex]
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\ln 4}{\ln1 - \ln 3}[/tex]
[tex]\implies \log_{\frac{1}{3}}(4)=\dfrac{\ln 4}{0 - \ln 3}[/tex]
[tex]\implies \log_{\frac{1}{3}}(4)=-\dfrac{\ln 4}{\ln 3}[/tex]
What type of number is -560,114?
Choose all answers that apply:
A Whole number
B
C
D
Integer
Rational
Irrational
Answer:
It is an integer.
It is rational.
Step-by-step explanation:
A whole number is in the set
0, 1, 2, 3...
This set starts at zero and only includes zero and the positive numbers. So the given number is NOT a whole number.
It IS an integer. Integers include positive and negative numbers. That is, whole numbers and their opposites.
The number IS rational. That is, it can be written as a ratio:
-560114/1
Since it is rational, it is not irrational.
Function A and Function B are linear functions.
Function A
Function B
y=x-1
Which statement is true?
The y-intercept of Function A is greater than the y-intercept of Function B.
The y-intercept of Function A is less than the y-intercept of Function B.
The y-intercept of Function A is greater than the y-intercept of Function B.
What is y-intercept ?
The y intercept of a graph is the point where the graph intersects the y-axis. We know that the x-coordinate of any point on the y-axis is 0. So the x-coordinate of a y-intercept is 0.
Here are the steps to find the y intercept of a function y = f(x),
we just substitute x = 0 in it.solve for y.represent the y-intercept as the point (0, y).Given:
Function A : y=x-1
y-intercept of Function A (x=0) is y = -1
y-intercept of Function B are -6 , -4 and -3.
Since , -1 > -6 , -1 > -4 and -1 > -3
Hence , the y-intercept of Function A is greater than the y-intercept of Function B.
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Compare these numbers.
>
<
=
none of the above
determine the z transform for each of the following sequences. sketch the pole-zero plot and indicate the region of convergence. indicate whether or not the (discrete time) fourier transform of the sequence exists.
The Fourier transform of this sequence does not exist
What is Fourier Transform?
The output of a Fourier transform is a function of frequency and is a mathematical transformation that breaks down functions into frequency components.
To determine the z-transform of a given sequence, we need to find the sum of the sequence's terms multiplied by z raised to the negative of the term's position. For example, given a sequence x[n], the z-transform X(z) is given by:
[tex]X(z) = sum_{n=-infinity}^{infinity} x[n] * z^{-n}[/tex]
To sketch the pole-zero plot of a z-transform, we need to plot the poles (values of z where the z-transform is infinite) and zeros (values of z where the z-transform is zero) in the complex plane. The region of convergence (ROC) is the set of values of z for which the sum defining the z-transform converges.
[tex]x[n] = a^n u[n][/tex], where a is a constant and u[n] is the unit step function[tex](u[n] = 1 for n > = 0[/tex] and [tex]u[n] = 0 for n < 0)[/tex]
The z-transform of this sequence is given by:
[tex]X(z) = sum_{n=0}^{infinity} a^n * z^{-n}= 1 / (1 - a*z^{-1})[/tex]
This z-transform has a pole at z = 1/a and a zero at z = 0. The pole-zero plot consists of a single pole at 1/a in the complex plane. The ROC is the set of values of z such that [tex]|a*z^{-1}| < 1[/tex], which is the region inside the unit circle centered at the origin. The Fourier transform of this sequence exists.
[tex]x[n] = (-1)^n u[n].[/tex]
The z-transform of this sequence is given by:
[tex]X(z) = sum_{n=0}^{infinity} (-1)^n * z^{-n}= 1 / (1 + z^{-1})[/tex]
This z-transform has a pole at z = -1 and a zero at z = 0. The pole-zero plot consists of a single pole at -1 in the complex plane. The ROC is the set of values of z such that [tex]|z^{-1}| < 1[/tex], which is the region inside the unit circle centered at the origin. The Fourier transform of this sequence exists.
[tex]x[n] = n^2 u[n].[/tex]
The z-transform of this sequence is given by:
[tex]X(z) = sum_{n=0}^{infinity} n^2 * z^{-n}= z^{-2} / (1 - z^{-1})^3[/tex]
This z-transform has poles at z = 1, z = 1, and z = 1. The pole-zero plot consists of three poles at the origin in the complex plane. The ROC is the set of values of z such that [tex]|z^{-1}| > 1[/tex], which is the region outside the unit circle centered at the origin.
Hence, The Fourier transform of this sequence does not exist.
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