Series

A series is an infinite ordered set of terms combined together by the additional rule.ex. (1/2+1/4+1/8+…) or (a1+a2+a3) .The term infinite series is sometimes used to emphasize the fact that series contain an infinite number of terms. The Riemann series theorem states that by a suitable arrangement of terms, a conditionally convergent series may be made to converge to any desired value. If the difference between successive terms of a series is constant, then the series is said to be an arithmetic series.

A series for which the ratio of two consecutive terms (a2+b2) is a constant function of the summation index 2 is called a geometric series. The more general case of the ratio, a rational function of 2 produces a series called a hyper geometric series. A series may be called divergent in the case that it may converge to a definite value.

Brun’s constant

Another well- known convergent infinite series is Brun’s constant. A special strong type of convergence is called uniform convergence and series which are uniformly convergent have a particular order. For example, the sum of a uniformly convergent series has a continuous function. A convergent series can be differentiated term by term, provided that the functions of the series have continuous derivatives and the series of derivatives is uniformly convergent.  

Semi- Convergence

A series is said to be semi- convergent if it is not absolutely convergent. Semi- convergent series was studied by Poisson (1823) who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem however came from Jacobi (1834) who attacked the question of the remainder from a different stand point and reached a different formula.

Generalizations

Asymptotic series

Asymptotic series otherwise known as asymptotic expansions are infinite series whose partial sums become good approximations in the limit of some point of the domain. In general they do not converge, but they are useful as sequence of approximations, each of which provides a value close to the desired answer for a finite number of terms. The difference is that an asymptotic series cannot be made to produce an answer as exact as desired.

Divergent series

A summability method is such an assignment of a limit to a subset of the set of divergent series which properly extends the classical notion of convergence. Under some circumstances, it is desirable to assign a limit to a series which fails to converge in the usual sense.

Questions:

• What is a Series in Mathematical terms?
• What is Asymptotic and Divergent series?