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In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a function whose domain is a directed set.The codomain of this function is usually some topological space.
The Fibonacci numbers were first described in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. [ 2 ] [ 3 ] [ 4 ] They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci , who introduced the sequence to Western ...
Thus, in this case the convergence of the rays transmitted by a lens is equal to the radius of the light source divided by its distance from the optics. This limits the size of an image or the minimum spot diameter that can be produced by any focusing optics, which is determined by the reciprocal of that equation; the divergence of the light ...
Unlike the preceding sequence, this sequence does not have any pattern that is easily discernible by inspection. Other examples are sequences of functions, whose elements are functions instead of numbers. The On-Line Encyclopedia of Integer Sequences comprises a large list of examples of integer sequences. [3]
In addition to the previously defined Q-linear convergence, a few other Q-convergence definitions exist. Given Definition 1 defined above, the sequence is said to converge Q-superlinearly to (i.e. faster than linearly) in all the cases where > and also the case =, =. [8]
The dominated convergence theorem applies also to measurable functions with values in a Banach space, with the dominating function still being non-negative and integrable as above. The assumption of convergence almost everywhere can be weakened to require only convergence in measure.
The geometric series + + + + … is an infinite series derived from a special type of sequence called a geometric progression, which is defined by just two parameters: the initial coefficient and the common ratio .
Many of these functions can be used to find their own solutions by repeatedly recycling the result back as input, but the rate of convergence can be slow, or the function can fail to converge at all, depending on the individual function. Steffensen's method accelerates this convergence, to make it quadratic.