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However, if the terms and their finite sums belong to a set that has limits, it may be possible to assign a value to a series, called the sum of the series. This value is the limit as n tends to infinity of the finite sums of the n first terms of the series if the limit exists. [9] [10] [11] These finite sums are called the partial sums of the
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.
where the left-hand side has to be interpreted as being the value obtained by using one of the aforementioned summation methods and not as the sum of an infinite series in its usual meaning. These methods have applications in other fields such as complex analysis , quantum field theory , and string theory .
e. In mathematics, a geometric series is a series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, the series is a geometric series with common ratio , which converges to the sum of . Each term in a geometric series is the geometric mean of the term before it ...
Calculus. In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: The first terms of the series sum to approximately , where is the natural logarithm and is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it ...
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 ...
t. e. In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point.
Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series.Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined.