0
Numerical Series Expansion
Mathematicians, scientists, and engineers often need solutions for difficult to unsolvable equations in order to progress in their work. Fortunately, through various methods of numerical and polynomial series expansions, some of the most problematic equations can be approximated to easily workable forms. Please post all questions regarding the various expansions and series, such as the Taylor and Maclaurin series expansions, the binomial expansion, and the geometric expansion, as well as the rules that govern them, into this category.
274 Questions
What is the formula to convert percent to liters?
Converting a percentage to liters is not a straightforward formula as percent is a measure of a part per hundred while liters are a unit of volume. To convert a percentage to liters, you would need to know the density of the substance in question. The formula would involve multiplying the volume of the substance by its density to determine the amount in liters.
What factors add up to 240 that add up to 140?
To find factors that add up to 240 and also add up to 140, we need to solve a system of equations. Let's represent the two factors as x and y. We have the equations x + y = 140 and x + y = 240. By solving this system, we find that the factors are 100 and 140.
What is the history of fourier series?
It is quite complicated, and starts before Fourier. Trigonometric series arose in problems connected with astronomy in the 1750s, and were tackled by Euler and others. In a different context, they arose in connection with a vibrating string (e.g. a violin string) and solutions of the wave equation.
Still in the 1750s, a controversy broke out as to what curves could be represented by trigonometric series and whether every solution to the wave equation could be represented as the sum of a trigonometric series; Daniel Bernoulli claimed that every solution could be so represented and Euler claimed that arbitrary curves could not necessarily be represented. The argument rumbled on for 20 years and dragged in other people, including Laplace. At that time the concepts were not available to settle the problem.
Fourier worked on the heat equation (controlling the diffusion of heat in solid bodies, for example the Earth) in the early part of the 19th century, including a major paper in 1811 and a book in 1822. Fourier had a broader notion of function than the 18th-century people, and also had more convincing examples.
Fourier's work was criticised at the time, and his insistence that discontinuous functions could be represented by trigonometric series contradicted a theorem in a textbook by the leading mathematician of the time, Cauchy.
Nonetheless Fourier was right; Cauchy (and Fourier, and everyone else at that time) was missing the idea of uniform convergence of a series of functions. Fourier's work was widely taken up, and also the outstanding problems (just which functions can be represented by Fourier series?; how different can two functions be if they have the same Fourier series?) were slowly solved.
Source: Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972, pages 478-481, 502-514, 671-678,and 964.
How do you convert a 1995 Escort 19 liter to a 1999 2 liter?
I believe it can be but there will be some sensors and other some componets from the 95 engine that will have to stay with the car, and be transferred to the 2 liter from the 1.9. The only real difference between a 1.9 and a 2.0 is the engine bore so it should work. The other differences will be year differences with sensor plugs, and possibly the ignition system.
What are the next set of numbers after 12 34 35 38 42?
Given any number it is easy to find a rule based on a polynomial of order 5+k such that the first five numbers are as listed in the question and the next k are the given "next" numbers.
However, use the rule
Un = (-6n4 + 83n3 - 411n2 + 874n - 468)/6
Accordingly, the next three numbers are 22, -71, -310
There is no next since numbers are infinitely dense. This means that there are infinitely many numbers between any two numbers. So, if you suggest that 84.1 is next, you are wrong, because there are infinitely many numbers between 84 and 84.1 and one of them has a better claim to be next. But then, there are infinitely many numbers between 84 and that number and one of these has an ever better claim to be next. And so on.
Therefore, there is no such thing as a next number.
When is a binomial distribution cumulative?
It is cumulative when you add together the probabilities of all events resulting in the given number or fewer successes.
What number when multiplied by 10 is 50 less than 650?
Let's call the number x.
When x is multiplied by 10, it is 50 less than 650. We can write this as:
10x+50 = 650
Now that we have an equation, let's solve it.
Firstly, subtract 50 from both sides
10x = 600
Then, divide both sides by 10
x = 60
The number 60 is 50 less than 650 when it is multiplied by 10.
Is this sequence 10 10.25 10.50625 10.76890625 arithmetic?
No, it is geometric, since each term is 1.025 times the previous.
An example of an arithmetic sequence would be 10, 10.25, 10.50, 10.75, 11.
Where can you find a site for selective school sample test papers?
In the UK, selective state schools use National Foundation for Educational Research (NFER) exams whereas independent selective schools exams are set by the Independent Schools Examinations Board (ISEB). Their web sites can offer some samples. More may be obtained from the sites for the individual schools.
What does the number 0408163245576318222527346277 mean?
It does not mean anything specifically. In a particular context it may have some meaning but you have chosen not to share that context with us.
What is converging and diverging?
Divergent and convergent are both boundaries that form different kinds of landmasses.
What is the proof of newton raphson iterative equation?
Suppose you have a differentiable function of x, f(x) and you are seeking the root of f(x): that is, a solution to f(x) = 0.Suppose x1 is the first approximation to the root, and suppose the exact root is at x = x1+h : that is f(x1+h) = 0.
Let f'(x) be the derivative of f(x) at x, then, by definition,
f'(x1) = limit, as h tends to 0, of {f(x1+h) - f(x1)}/h
then, since f(x1+h) = 0, f'(x1) = -f(x1)/h [approx] or h = -f'(x1)/f(x1) [approx]
and so a better estimate of the root is x2 = x1 + h = x1 - f'(x1)/f(x1).
What is the formula for exponential growth?
It can vary , but it is something along the lines of
G(t) = Ae^(xt)
Where
'G' is growth at time 't'
'A' is a constnt
'e' is the exponential 2.7818....
'x' is the variable factor
't' is the time.
e^(xt) is the exponential raised to the power of 'variable factor multiplied to time'.
What is the Frequency of a free falling object?
The frequency of a free falling object is determined by the rate at which it falls due to gravity. This rate is typically constant (9.8 m/s^2 on Earth) and results in a consistent frequency of oscillation for the object as it free falls.
What is meant by Cauchy's constants?
The refractive index of a substance can be expanded out with a Fourier transform into the Cauchy equation n = A + B/λ2 + C/λ4 where n is the refractive index and λ is the wavelength of the electromagnetic wave in question. The coefficients in this equation, A, B, and C, are called the Cauchy constants and can be figured out experimentally.
What is a example of convergence?
An example of convergence is when a group of friends with differing opinions gradually come to a consensus on where to go for dinner by considering everyone's preferences and finding a common choice that satisfies all parties.
The natural frequencies of an object are the frequencies at which the object tends to vibrate easily. Harmonics are frequencies that are integer multiples of the fundamental frequency. When an object is excited at its natural frequencies, it tends to resonate and produce harmonics of those frequencies.
