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List of mathematical series. This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. is a Bernoulli polynomial. is an Euler number. is the Riemann zeta function. is the gamma function. is a polygamma function. is a polylogarithm.
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, [1] and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. [2] Since the problem had withstood the attacks of ...
In mathematics, a series is, roughly speaking, an addition of infinitely many quantities, one after the other. [1] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions.
A geometric series is a unit series, meaning the series sum converges to one, if and only if |r| < 1 and a + r = 1. The terms of a geometric series are also the elements of a generalized Fibonacci sequence (a recursively defined sequence with F n = F n-1 + F n-2 ) when the series's common ratio r satisfies the constraint 1 + r = r 2 , which is ...
A summation method that is linear and stable cannot sum the series 1 + 2 + 3 + ⋯ to any finite value. (Stable means that adding a term at the beginning of the series increases the sum by the value of the added term.) This can be seen as follows. If + + + =, then adding 0 to both sides gives
Euler summation. In the mathematics of convergent and divergent series, Euler summation is a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σ an, if its Euler transform converges to a sum, then that sum is called the Euler sum ...
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 geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying ...
The previous formula becomes. {\displaystyle \sum _ {1\leq n\leq x}a_ {n}\phi (n)=A (x)\phi (x)-\int _ {1}^ {x}A (u)\phi ' (u)\,du.} A common way to apply Abel's summation formula is to take the limit of one of these formulas as . The resulting formulas are. These equations hold whenever both limits on the right-hand side exist and are finite ...