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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
What is the sum of the infinite geometric series?
The sum of the series
a + ar + ar2 + ... is a/(1 - r) for |r| < 1
How can you tell if a infinite geometric series has a sum or not?
The geometric series is, itself, a sum of a geometric progression. The sum of an infinite geometric sequence exists if the common ratio has an absolute value which is less than 1, and not if it is 1 or greater.
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
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).
This site no longer allows me to enter subscripts so I will use brackets: a(n) to indicate the nth term.
a(n) = a(1) + (n-1)*d where d is the common difference between the terms of the arithmetic sequence.
Therefore, d = [a(n) - a(1)]/(n-1)
Then, the appropriate arithmetic series is
S(n) = 1/2*n[2*a(1) + (n-1)*d] where all the terms on the right hand side are known.
Does a sum of infinite ones equal a sum of infinite twos?
Yes, the sum of infinite ones equal the sum of infinite twos.
Yes. A cubic equation can have 3 real roots. Depending on their size, each of three intervals could contain a root. In that case different intervals must give different roots.
Yes. A cubic equation can have 3 real roots. Depending on their size, each of three intervals could contain a root. In that case different intervals must give different roots.
Yes. A cubic equation can have 3 real roots. Depending on their size, each of three intervals could contain a root. In that case different intervals must give different roots.
Yes. A cubic equation can have 3 real roots. Depending on their size, each of three intervals could contain a root. In that case different intervals must give different roots.
1,424,007,025.076
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
What is the relationship between the binomial expansion and binomial distribution?
First i will explain the binomial expansion
Interval estimation of the parameters of poisson distribution?
n order to fit the Poisson distribution, we must estimate a value for λ from the observed data. Since the average count in a 10-second interval was 8.392, we take this as an estimate of λ (recall that the E(X) = λ) and denote it by ˆλ.
That is the correct spelling of "fourths" (meaning the plural of one-fourth).
What does culture convergence mean?
Cultural convergence is when the similarities between different cultures in the same nation are driven by global and common brands. The similarities are with aspects like consumer preferences, beliefs, aspirations, and values.
9!
~
What is the difference between an arithmetic series and a geometric series?
An arithmetic series is the sequence of partial sums of an arithmetic sequence. That is, if A = {a, a+d, a+2d, ..., a+(n-1)d, ... } then the terms of the arithmetic series, S(n), are the sums of the first n terms and S(n) = n/2*[2a + (n-1)d]. Arithmetic series can never converge.
A geometric series is the sequence of partial sums of a geometric sequence. That is, if G = {a, ar, ar^2, ..., ar^(n-1), ... } then the terms of the geometric series, T(n), are the sums of the first n terms and T(n) = a*(1 - r^n)/(1 - r). If |r| < 1 then T(n) tends to 1/(1 - r) as n tends to infinity.
How do you write 543201065483?
Five hundred forty-three billion, two hundred one million, sixty-five thousand, four hundred eighty-three.
