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Series Solutions: Preliminaries (A Brief Review of Infinite Series, Power Series and a Little Complex Variables)
Published in Kenneth B. Howell, Ordinary Differential Equations, 2019
If a series converges but is not absolutely convergent, then it is converging because each term “cancels out” some of the previous terms, and the series is said to be conditionally convergent. Such a convergence is somewhat unstable and can be upset by, say, rearranging the terms of the series in a clever way. Because of this, we will much prefer series that converge absolutely.
Infinite series
Published in C.W. Evans, Engineering Mathematics, 2019
Any series ∑an, real or complex, which has the property that ∑|an| converges is called an absolutely convergent series. This test tells us that if a series is absolutely convergent then it is convergent. There are many series which are convergent but which are not absolutely convergent. These series are called conditionally convergent.
The algebra of bounded operators on a Banach space
Published in Orr Moshe Shalit, A First Course in Functional Analysis, 2017
It is also clear that this limit does not depend on the order in which the sum is taken. Therefore, this sum also exists in the same sense as when discussing convergence of infinite series in Hilbert spaces. Making the obvious modification of Definition 3.3.4 to normed spaces, we can say that in a Banach space, an absolutely convergent series is also convergent.
On the abs-polynomial expansion of piecewise smooth functions
Published in Optimization Methods and Software, 2021
A. Griewank, T. Streubel, C. Tischendorf
To show this we proceed again by induction on the instruction index i. For the independent variables the expansions are finite so certainly absolutely convergent with . For linear operations and the multiplication between some and we get the minimal convergence radius . The latter follows from the fact that the Cauchy product of absolutely convergent series is also absolutely convergent to the correct limit. For the absolute value operation , we find that simply . Now let us consider finally a univariate intrinsic . Provided ϕ is real analytic near it has a positive convergence radius r>0. Then we can restrict such that and the series for is absolutely convergent. Consequently for the series on the right-hand side of (11) is absolutely convergent and can by Fubini's principle be bracketed to yield the abs-polynomial series expansion of . Thus, we can set to complete the proof.
Admissible inertial manifolds for neutral equations and applications
Published in Dynamical Systems, 2021
Thi Ngoc Ha Vu, Thieu Huy Nguyen, Anh Minh Le
Let now and be two successive and distinct eigenvalues with , and let P be the orthogonal projection onto the first N eigenvectors of the operator A. Denote by the semigroup generated by . Since is of finite dimension, the restriction of to can be extended to the whole line . For we then recall the following dichotomy estimates (see [7,23]): and We then define the Green's function as follows: It is obvious that maps X into . Also, by the dichotomy estimates (3) and for we have where and Moreover, one can see that is a Banach space endowed with the norm and let E be admissible function space and be its associate space defined as in Definition 1.3. Then, we set where the notation is denoted for the function . Now, is endowed with the norm It can be seen that is a Banach space [22]. Actually, the result in [22] is more general, namely is a Banach space for any Banach space X instead of . The proof is based on the facts that X is a Banach space, E is a Banach lattice, and that a normed space is complete if and only if every absolutely convergent series in that space is convergent.