Dec 3, 2025 · A geometric series sum_ (k)a_k is a series for which the ratio of each two consecutive terms a_ (k+1)/a_k is a constant function of the summation index k. The more …

A hypergeometric series sum_ (k)c_k is a series for which c_0=1 and the ratio of consecutive terms is a rational function of the summation index k, i.e., one for which (c_ (k+1))/ (c_k)= (P …

Dec 3, 2025 · If the difference between successive terms of a series is a constant, then the series is said to be an arithmetic series. A series for which the ratio of each two consecutive terms is …

A sum is the result of an addition. For example, adding 1, 2, 3, and 4 gives the sum 10, written 1+2+3+4=10. (1) The numbers being summed are called addends, or sometimes summands. …

Dec 3, 2025 · The analytic summation of a hypergeometric series. Powerful general techniques of hypergeometric summation include Gosper's algorithm, Sister Celine's method, Wilf-Zeilberger …

Dec 3, 2025 · The series sum_ (k=1)^inftykr^k=sum_ (k=1)^inftysum_ (i=k)^inftyr^i=r/ ( (1-r)^2), valid for 0<r<1.

The function is often called "the" hypergeometric function or Gauss's hypergeometric function, and is implemented in the Wolfram Language as Hypergeometric2F1 [a, b, c, x].

Dec 3, 2025 · The series in this theorem is called an SU (n) series (Milne 1985; Bhatnagar 1995, p. 22). Many other q-results, including the q-binomial theorem and q-Saalschütz sum, can be …

A double sum is a series having terms depending on two indices, sum_ (i,j)b_ (ij). (1) A finite double series can be written as a product of series sum_ (i=1)^ (m)sum_ (j=1)^ (n)x_iy_j = …

Dec 3, 2025 · An arithmetic series is the sum of a sequence {a_k}, k=1, 2, ..., in which each term is computed from the previous one by adding (or subtracting) a constant d. Therefore, for k>1, …