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The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need for parentheses, and the result is the same irrespective of the order of the summands ...
The technique of the previous example may also be applied to other Dirichlet series. If a n = μ ( n ) {\displaystyle a_{n}=\mu (n)} is the Möbius function and ϕ ( x ) = x − s {\displaystyle \phi (x)=x^{-s}} , then A ( x ) = M ( x ) = ∑ n ≤ x μ ( n ) {\displaystyle A(x)=M(x)=\sum _{n\leq x}\mu (n)} is Mertens function and
The example function has an easy-to-find anti-derivative so estimating the integral by Riemann sums is mostly an academic exercise; however it must be remembered that not all functions have anti-derivatives so estimating their integrals by summation is practically important.
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, is taken to have the value
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.
This sum can be found quickly by taking the number n of terms being added (here 5), multiplying by the sum of the first and last number in the progression (here 2 + 14 = 16), and dividing by 2: (+) In the case above, this gives the equation:
A summation equation compares to a difference equation as an integral equation compares to a differential equation. The Volterra summation equation is: = + = (,, ()) where x is the unknown function, s, a, t are integers, and f, k are known functions.
For example, many summation methods are used in mathematics to assign numerical values even to a divergent series. In particular, the methods of zeta function regularization and Ramanujan summation assign the series a value of − + 1 / 12 , which is expressed by a famous formula: [2] + + + + =,