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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
Proof of finite difference method using Taylor series?
The finite difference method approximates derivatives using Taylor series expansions. For example, the first derivative ( f'(x) ) can be expressed as ( f'(x) = \frac{f(x+h) - f(x)}{h} + O(h) ), where ( O(h) ) represents the error term. By expanding ( f(x+h) ) using Taylor series, we can isolate and approximate the derivative, demonstrating that the method converges to the true derivative as the step size ( h ) approaches zero. This approach can similarly be applied to higher-order derivatives and different difference schemes.
How do you spell this number 180000?
The number 180000 (180,000) is spelled "one hundred and eighty thousand".
(The alternate is "one hundred eighty thousand" which may be less clear without hyphenation.)
What is binomial expansion theorem?
We often come across the algebraic identity (a + b)2 = a2 + 2ab + b2. In expansions of smaller powers of a binomial expressions, it may be easy to actually calculate by working out the actual product. But with higher powers the work becomes very cumbersome.
The binomial expansion theorem is a ready made formula to find the expansion of higher powers of a binomial expression.
Let ( a + b) be a general binomial expression. The binomial expansion theorem states that if the expression is raised to the power of a positive integer n, then,
(a + b)n = nC0an + nC1an-1 b+ nC2an-2 b2+ + nC3an-3 b3+ ………+ nCn-1abn-1+ + nCnbn
The coefficients in each term are called as binomial coefficients and are represented in combination formula. In general the value of the coefficient
nCr = n!r!(n-r)!
It may be interesting to note that there is a pattern in the binomial expansion, related to the binomial coefficients. The binomial coefficients at the same position from either end are equal. That is,
nC0 = nCn nC1 = nCn-1 nC2 = nCn-2 and so on.
The advantage of the binomial expansion theorem is any term in between can be figured out without even actually expanding.
Since in the binomial expansion the exponent of b is 0 in the first term, the general term, term is defined as the (r+1)th b term and is given by Tr+1 = nCran-rbr
The middle term of a binomial expansion is [(n/2) + 1]th term if n is even. If n is odd, then terewill be two middle terms which are [(n+1)/2]th and [(n+3)/2]th terms.
Sounds pretty sexy, eh? See link. http://en.wikipedia.org/wiki/Binomial_expansion
Program to find the sum of harmonic series?
Let's assume that you want the sum of the general harmonic series:
sum(n=0,inf): 1/(an+b)
Since we know that the harmonic series will converge to infinity, we'll also assume that you want the sum from 0 to n.
double genHarmonic(const double n, const double a, const double b) {
double sum = 0.0;
// perform calculations
int k;
for(k = 0; k <= n; ++k) {
sum += 1.0 / (a * k + b);
}
return sum;
}
What is a harmonic convergence?
Harmonic convergence is when the two spirit portals connect from the north and south poles in the spirit world. It is known to have a tree of time which can bring you back to any of the memories in your life. The ancients would meditate under the tree and connect with the great cosmic energy.
Is Fourier series used in power electronics to generate harmonics?
Yes, Fourier series can be used in power electronics to analyze and generate harmonics. By decomposing complex waveforms into their sinusoidal components, engineers can understand and manipulate the harmonic content of signals. This is particularly useful in applications like pulse-width modulation (PWM) for inverters, where controlling harmonics is essential for improving power quality and efficiency. Additionally, Fourier analysis helps in designing filters to mitigate unwanted harmonics in power systems.
What would be a coefficients of the expansion of the Pascal's Triangle?
its so simple....
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
.........................................
that is pascal's triangle coeffient law......just sum the two numbers and answer is yours
What is the sum of a 6-term geometric series if the first term is 21 and the last term is 352947?
U1 = a = 21
U6 = ar5 = 352947
where Un is the nth term, with first term, a, and commn ratio r.
Dividing the second equation by the first, r5 = 16807
Taking the fifth root, r = 7.
The S6 = a*(r6 - 1)/(r - 1) = 21*(76-1)/(7-1) = 411768
What properties of a function can be discovered from its Maclaurin series?
Yes. If the Maclaurin expansion of a function locally converges to the function, then you know the function is smooth. In addition, if the residual of the Maclaurin expansion converges to 0, the function is analytic.
How is poisson distribution related to binomial distribution?
The Poisson distribution is a limiting case of the binomial distribution when the number of trials is very large and the probability of success is very small. The Poisson distribution is used to model the number of occurrences of rare events in a fixed interval of time or space, while the binomial distribution is used to model the number of successful outcomes in a fixed number of trials.
What is harmonic as applied to fourier series?
When we do a Fourier transformation of a function we get the primary term which is the fundamental frequency and amplitude of the Fourier series. All the other terms, with higher frequencies and lower amplitudes, are the harmonics.
How do you derive Moment generating function of Pareto distribution?
The moment generating function for any real valued probability distribution is the expected value of e^tX provided that the expectation exists.
For the Type I Pareto distribution with tail index a, this is
a*[-x(m)t)^a*Gamma[-a, -x(m)t)] for t < 0, where x(m) is the scale parameter and represents the least possible positive value of X.
What will going tofollow to pentillions?
A pentillion was the name proposed for what is not known as a quadrillion = 10^15.
This was necessitated by the confusion between the US and European use of billion (thousand million or million million) and higher numbers. New names, based on Greek roots, were proposed. In many countries the European names have now been dropped in favour of the US names.
So the next term would have been a hextillion.
What does it mean to make a conjuncture?
Conjuncture is basically a joining of objects words or phrases. So to make a conjuncture would be to join things together. In propositional calculus making a conjuncture would mean joining a string of statements together.
Example: given the two statements
IF All A's are letters;
And All letters are are in the alphabet;
then the Conjuncture would be
All A's are in the alphabet.
