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In mathematics, a telescoping series is a series whose general term is of the form = +, i.e. the difference of two consecutive terms of a sequence (). As a consequence the partial sums of the series only consists of two terms of ( a n ) {\displaystyle (a_{n})} after cancellation.
The commented Poisson problem does not have a solution for any functional boundary conditions f 1, f 2, g 1, g 2; however, given f 1, f 2 it is always possible to find boundary functions g 1 *, g 2 * so close to g 1, g 2 as desired (in the weak convergence meaning) for which the problem has solution. This property makes it possible to solve ...
If an equation can be put into the form f(x) = x, and a solution x is an attractive fixed point of the function f, then one may begin with a point x 1 in the basin of attraction of x, and let x n+1 = f(x n) for n ≥ 1, and the sequence {x n} n ≥ 1 will converge to the solution x.
More sophisticated types of convergence of a series of functions can also be defined. In measure theory, for instance, a series of functions converges almost everywhere if it converges pointwise except on a set of measure zero. Other modes of convergence depend on a different metric space structure on the space of functions under consideration.
The Riemann zeta function is defined for real > by the convergent series = = = + + +, which for = would be the harmonic series. It can be extended by analytic continuation to a holomorphic function on all complex numbers except x = 1 {\displaystyle x=1} , where the extended function has a simple pole .
An arithmetico-geometric series is a sum of terms that are the elements of an arithmetico-geometric sequence. Arithmetico-geometric sequences and series arise in various applications, such as the computation of expected values in probability theory , especially in Bernoulli processes .
The image of a function f(x 1, x 2, …, x n) is the set of all values of f when the n-tuple (x 1, x 2, …, x n) runs in the whole domain of f.For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value.
The function f is convex (it is a maximum of linear functions). Denote the minimum value by f*. Then the answer to the decision problem is "yes" iff f*≤0. Step 4: In the optimization problem min z f(z), we can assume that z is in a box of side-length 2 L, where L is the bit length of the problem data.