Concept
Series (mathematics)
Related concepts (65)
James Gregory (mathematician)
James Gregory FRS (November 1638 – October 1675) was a Scottish mathematician and astronomer. His surname is sometimes spelt as Gregorie, the original Scottish spelling. He described an early practical design for the reflecting telescope – the Gregorian telescope – and made advances in trigonometry, discovering infinite series representations for several trigonometric functions.
Banach limit
In mathematical analysis, a Banach limit is a continuous linear functional defined on the Banach space of all bounded complex-valued sequences such that for all sequences , in , and complex numbers : (linearity); if for all , then (positivity); where is the shift operator defined by (shift-invariance); if is a convergent sequence, then . Hence, is an extension of the continuous functional where is the complex vector space of all sequences which converge to a (usual) limit in .
Actual infinity
In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These might include the set of natural numbers, extended real numbers, transfinite numbers, or even an infinite sequence of rational numbers. Actual infinity is to be contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps.
Abel's test
In mathematics, Abel's test (also known as Abel's criterion) is a method of testing for the convergence of an infinite series. The test is named after mathematician Niels Henrik Abel. There are two slightly different versions of Abel's test – one is used with series of real numbers, and the other is used with power series in complex analysis. Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions dependent on parameters.
Linear approximation
In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations. Given a twice continuously differentiable function of one real variable, Taylor's theorem for the case states that where is the remainder term.