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If α is a nonnegative integer n, then all terms with k > n are zero, [5] and the infinite series becomes a finite sum, thereby recovering the binomial formula. However, for other values of α , including negative integers and rational numbers, the series is really infinite.
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 mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total.Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
Faulhaber's formula concerns expressing the sum of the p-th powers of the first n positive integers = = + + + + as a (p + 1)th-degree polynomial function of n. The first few examples are well known. For p = 0, we have ∑ k = 1 n k 0 = ∑ k = 1 n 1 = n . {\displaystyle \sum _{k=1}^{n}k^{0}=\sum _{k=1}^{n}1=n.}
In number theory, the totient summatory function is a summatory function of Euler's totient function defined by Φ ( n ) := ∑ k = 1 n φ ( k ) , n ∈ N . {\displaystyle \Phi (n):=\sum _{k=1}^{n}\varphi (k),\quad n\in \mathbb {N} .}
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation , named after Niels Henrik Abel who introduced it in 1826.
for k = 0, 1, 2, ..., n, where =!! ()! is the binomial coefficient. The formula can be understood as follows: p k q n−k is the probability of obtaining the sequence of n independent Bernoulli trials in which k trials are "successes" and the remaining n − k trials