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t. e. In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity. [ 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 ...
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.
Ant on a rubber rope. The ant on a rubber rope is a mathematical puzzle with a solution that appears counterintuitive or paradoxical. It is sometimes given as a worm, or inchworm, on a rubber or elastic band, but the principles of the puzzle remain the same. The details of the puzzle can vary, [1] [2] but a typical form is as follows:
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 ...
In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence defines a series S that is denoted. The n th partial sum Sn is the sum of the first n terms of the sequence; that is, A series is convergent (or converges) if and only if the sequence of its partial sums tends to a limit ...
Power series. In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the n th term and c is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions.
N. H. Abel, letter to Holmboe, January 1826, reprinted in volume 2 of his collected papers. In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms ...
Calculus. In mathematics, an alternating series is an infinite series of the form or with an > 0 for all n. The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges .