Ad
related to: finite sum calculator average value formula excel download pdf version
Search results
Results From The WOW.Com Content Network
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
Very similar to the second example above, let (X n) n∈ be a sequence of independent, symmetric random variables, where X n takes each of the values 2 n and –2 n with probability 1 / 2 . Let N be the first n ∈ such that X n = 2 n. Then, as above, N has finite expectation, hence assumption holds.
Note that this approximation corresponds to a finite volume method representation whereby the discrete values represent averages of the state variables over the cells. Exact relations for the averaged cell values can be obtained from the integral conservation laws. Obtain the solution for the local Riemann problem at the cell interfaces. This ...
It is used to prove Kronecker's lemma, which in turn, is used to prove a version of the strong law of large numbers under variance constraints. It may be used to prove Nicomachus's theorem that the sum of the first cubes equals the square of the sum of the first positive integers. [2]
The algorithm performs summation with two accumulators: sum holds the sum, and c accumulates the parts not assimilated into sum, to nudge the low-order part of sum the next time around. Thus the summation proceeds with "guard digits" in c , which is better than not having any, but is not as good as performing the calculations with double the ...
If the sum is of the form = ()where ƒ is a smooth function, we could use the Euler–Maclaurin formula to convert the series into an integral, plus some corrections involving derivatives of S(x), then for large values of a you could use "stationary phase" method to calculate the integral and give an approximate evaluation of the sum.
In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus .
Final value theorems for obtaining () have applications in probability and statistics to calculate the moments of a random variable. Let R ( x ) {\displaystyle R(x)} be cumulative distribution function of a continuous random variable X {\displaystyle X} and let ρ ( s ) {\displaystyle \rho (s)} be the Laplace–Stieltjes transform of R ( x ...