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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.
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.
There are many other combinatorial interpretations of binomial coefficients (counting problems for which the answer is given by a binomial coefficient expression), for instance the number of words formed of n bits (digits 0 or 1) whose sum is k is given by (), while the number of ways to write = + + + where every a i is a nonnegative integer is ...
The Bernoulli numbers have various definitions (see Bernoulli number#Definitions), such as that they are the coefficients of the exponential generating function = ( +) = =!. Then Faulhaber's formula is that ∑ k = 1 n k p = 1 p + 1 ∑ k = 0 p ( p + 1 k ) B k n p − k + 1 . {\displaystyle \sum _{k=1}^{n}k^{p}={\frac {1}{p+1}}\sum _{k=0}^{p ...
The sum is taken over all combinations of nonnegative integer indices k 1 through k m such that the sum of all k i is n. That is, for each term in the expansion, the exponents of the x i must add up to n. [2] [a] In the case m = 2, this statement reduces to that of the binomial theorem. [2]
For k ≤ np, upper bounds can be derived for the lower tail of the cumulative distribution function (;,) = (), the probability that there are at most k successes. Since Pr ( X ≥ k ) = F ( n − k ; n , 1 − p ) {\displaystyle \Pr(X\geq k)=F(n-k;n,1-p)} , these bounds can also be seen as bounds for the upper tail of the cumulative ...
The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series. This special case of a matrix summability method is named for the Italian analyst Ernesto Cesàro (1859–1906).
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.