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Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted c and c 0, with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called an FK-space.
The name "Fibonacci sequence" was first used by the 19th-century number theorist Édouard Lucas. [21] Solution to Fibonacci rabbit problem: In a growing idealized population, the number of rabbit pairs form the Fibonacci sequence.
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. For instance, the sequence
Short exact sequences are exact sequences of the form 0 → A → f B → g C → 0. {\displaystyle 0\to A\xrightarrow {f} B\xrightarrow {g} C\to 0.} As established above, for any such short exact sequence, f is a monomorphism and g is an epimorphism.
For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with a common ratio of 1/2. Examples of a geometric sequence are powers r k of a fixed non-zero number r, such as 2 k and 3 k. The general form of a geometric sequence is
Find recurrence relations for sequences—the form of a generating function may suggest a recurrence formula. Find relationships between sequences—if the generating functions of two sequences have a similar form, then the sequences themselves may be related. Explore the asymptotic behaviour of sequences. Prove identities involving sequences.
Lucas sequences are used in probabilistic Lucas pseudoprime tests, which are part of the commonly used Baillie–PSW primality test. Lucas sequences are used in some primality proof methods, including the Lucas–Lehmer–Riesel test, and the N+1 and hybrid N−1/N+1 methods such as those in Brillhart-Lehmer-Selfridge 1975. [4]
For example, x, y(1-x), z(1-x) is a regular sequence in the polynomial ring C[x, y, z], while y(1-x), z(1-x), x is not a regular sequence. But if R is a Noetherian local ring and the elements r i are in the maximal ideal, or if R is a graded ring and the r i are homogeneous of positive degree, then any permutation of a regular sequence is a ...