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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 ...
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.
These equations describe boundary-value problems, in which the solution-function's values are specified on boundary of a domain; the problem is to compute a solution also on its interior. Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences. [2 ...
A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one variable, is a horizontal line. In this context, a function that is also a linear map (the other meaning) may be referred to as a homogeneous linear function or a linear form.
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 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.