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An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
In general, grouping the terms of a series creates a new series with a sequence of partial sums that is a subsequence of the partial sums of the original series. This means that if the original series converges, so does the new series after grouping: all infinite subsequences of a convergent sequence also converge to the same limit.
In mathematics, a sequence is a list of objects (or events) which have been ordered in a sequential fashion; such that each member either comes before, or after, every other member. More formally, a sequence is a function with a domain equal to the set of positive integers. A series is a sum of a sequence of terms. That is, a series is a list ...
Nevertheless, holonomic sequences play an important role in various areas of mathematics. For example, many special functions have a Taylor series whose sequence of coefficients is holonomic. The use of the recurrence relation allows a fast computation of values of such special functions. Not all sequences can be specified by a recurrence relation.
Recamán's sequence: 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, ... "subtract if possible, otherwise add": a(0) = 0; for n > 0, a(n) = a(n − 1) − n if that number is positive and not already in the sequence, otherwise a(n) = a(n − 1) + n, whether or not that number is already in the sequence. A005132: Look-and ...
This last non-simple continued fraction (sequence A110185 in the OEIS), equivalent to = [;,,,,,...], has a quicker convergence rate compared to Euler's continued fraction formula [clarification needed] and is a special case of a general formula for the exponential function:
Diagram illustrating three basic geometric sequences of the pattern 1(r n−1) up to 6 iterations deep.The first block is a unit block and the dashed line represents the infinite sum of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3 respectively.
A negative-order reversal of this sequence powers formula corresponding to the operation of repeated integration is defined by the zeta series transformation and its generalizations defined as a derivative-based transformation of generating functions, or alternately termwise by and performing an integral transformation on the sequence ...