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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
The gamma function is an extension of the concept of a factorial. For positive integers n, Gamma(n) = (n - 1)!The function is defined for all complex numbers z for which the real part of z is positive, and it is the integral, from 0 to infinity of [x^(z-1) * e^(-x) with respect to x.
Condition for an infinite geometric series with common ratio to be convergent?
The absolute value of the common ratio is less than 1.
Is the Poisson probability distribution discrete or continuous?
The Poisson distribution is discrete.
Pascal's triangle how it it used in binomial expansion?
You can find the coefficients of an expanded binomial using the numbers in Pascal's triangle. 1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1 These are a few rows of Pascals triangle. Now let's look at a few binomials, expanded to the second and third powers. (a+b)2=a2 +2(ab) + b2 notice the coefficients are the numbers in the second row of the triangle above. (a+b)3= a3+3(a2b)+3(ab2)+b3 and once again note that the coefficients are the numberin the third line of Pascal's triangle. The first line, by the way, which is 1,1 is the coefficient of (a+b)1 This will work for any power of the binomial. There are generalized form for non-integer powers.
let binomial be (a + b)now (a+b)3 will be (a+b)(a+b)2
= (a+b)(a2 + 2ab+ b2)
= a(a2+ 2ab+ b2) + b(a2 + 2ab+ b2)
= a3+ 2a2b+ ab2 + a2b + 2ab2 + b3
= a3+ 2a2b+ ab2 + a2b + 2ab2 + b3
= a3 +3a2b + 3ab2 +b3
hope it helped... :D
What is a binomial distribution?
The binomial distribution is one in which you have repeated trials of an experiment in which the outcomes of the experiment are independent, the probability of the outcome is constant.
If there are n trials and the probability of "success" in each trail is p, then the probability of exactly r successes is (nCr)*p^r*(1-p)^(n-r) :
where nCr = n!/[r!*(n-r)!]
and n! = n*(n-1)*...*3*2*1
For a binomial distribution with n15 as p changes from .50 toward .05 the distribution will become?
with n=15 as fixed, as p=0.5 changes to p=.05 the binmomial distribution will shift to the right.
How do you solve geometric sequence and series?
There can be no solution to geometric sequences and series: only to specific questions about them.
It is 58465.
What is the method to calculate cp and cpk for unilateral tolerance?
Calculation
a) Standard Deviation (s) is calculated as follows:
√∑(X - Xi)2
n-1
For bilateral tolerances:
i) Capability Index (Cp) = USL - LSL / 6s
ii) Performance Index or Centering of Process (CmK) =
Minimum of USL- X or X - LSL
3s 3s
For single sided tolerances:
i) Performance Index or Centering of Process (CpK) =
a) Find out minimum observed value from the data values.
b) Find out Z = X - Minimum Observed Value
s
c) Corresponding to the arrived value of 'Z' above, choose the value of 'k' from Table given in Annexure-I.
d) Find out CmK = USL - X or X - LSL
ks ks
as the case may be.
USL for runout, roundness, surface finish etc.
LSL for Min. Hardness
M.Ananthakrishnan
Is it true when a sequence is divergent then its subsequences are divergent explain?
Not always true. Eg the divergent series 1,0,2,0,3,0,4,... has both convergent and divergent sub-sequences.
How does an infinite geometric series apply to being pushed on a swing?
Probably the movement on a swing can be approximated by assuming that the magnitude of each swing will be a certain percentage of the previous swing (because of lost energy).
Rapidity of convergence is a relative concept whose meaning comes from the "comparison test" for convergence. One version of the comparison test says that if (an) is convergent and abs(bn)<K*abs(an) for all n, for some K>0, then (bn) must be convergent too.
How does this map illustrate cultural convergence?
Answer this question…
The spread of coffee drinking to many parts of the world increases the similarity between cultures.
It is when an operation is wrong and you have to find the mistake and correct and get the right answer
Does a sum of infinite ones equal a sum of infinite twos?
Yes, the sum of infinite ones equal the sum of infinite twos.
