Infinite sequences can be described by a formula, such as the Fibonacci sequence, which starts with 0, 1 and each subsequent element is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, … A finite sequence has a fixed number of elements, while an infinite sequence continues indefinitely. The elements of a sequence are arranged in a specific order, and each element can be identified by its position in the sequence.įor example, the sequence of even numbers can be written as follows: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, … In this sequence, each element is obtained by adding 2 to the previous element. What is a Sequence?Ī sequence is a set of numbers or other objects that are arranged in a particular order, usually based on a pattern or rule. They also have practical applications in various fields, including engineering, physics, and computer science. Sequences and series help us to understand patterns, relationships, and mathematical structures. Arithmetic operations with power series are discussed.In mathematics, a sequence, and a series are two related concepts that are often used in various branches of mathematics, including calculus, number theory, and algebra. We define the Taylor series of an infinitely differentiable function, and give sufficient conditions for the Taylor series to converge to the function on some interval. It is shown that a power series that converges on an open interval defines an infinitely differentiable function on that interval. We give sufficient conditions for the limit of a sequence of functions or the sum of an infinite series of functions to be continuous, integrable, or differentiable. Cauchy’s uniform convergence criteria for sequences and series are proved, as is Dirichlet’s test for uniform convergence of a series.
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