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Infinite series

The idea of an infinite series has a long history, and one of the impetus for its formal definition started in process of  approximations of functions in form of power series like expansions. The first attempts can be found in the work of Madhava of Sangamagrama   in India in the 14th century who found  the first terms of the "Maclaurin series" for some trigonometric functions. For instance, he discovered the series expansion

and used it to calculate .

In 1667 James Gregory    issued his Vera Circuli et Hyperbolae Quadratura, in which he showed how the areas of the circle and hyperbola can be computed using infinite convergent series. The book also contains series expansions of trigonometric function sin, cos, arcsin and arccos. Several decades later and independently Colin Maclaurin     began to work with infinite series and published several what we now call Maclaurin series expansion in his Methodus incrementorum directa et inversa.  However, his expansions were published by James Gregory, but Maclaurin wasn't aware of this. In 1715 Brook Taylor derived the Taylor series, as a limit of Taylor polynomials, for all fundamental elementary functions. In 18th century Leonhard Euler developed the theory of hypergeometric series and q-series. Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even if they were not convergent. From our point of view the modern theory of the theory  of infinite series began with C.F.Gauß in the 19th century.  His memoir (1812) on hypergeometric series  contains simpler convergence criteria. He also opens here the problems of remainders and the range of convergence. The formal theory of infinite series started with A.L.Cauchy in 1821.  Although already Euler and Gauß had given come convergence criteria, it was Cauchy who insisted on  tests of convergence.  Cauchy advanced the theory of power series by extending the subject by power series expansion of a complex function.

A formal definition of an infinite series starts with an infinite sequences , of complex numbers. The sum of an infinite series

(1)

is defined as the limit of the sequence of partial sums

(2)

as , if that limit exists. If exists and is finite, the series (1) is said to converge.  If this limit is infinite or does not exist, the series (1) is said to be divergent. In some situation it is necessary to distinguish between these later cases. If the limit is , or , we say that the series diverges to , or , respectively.  If the limit does not exist, the series is said to oscillate.

The elements of sequence are called the terms of the series (1)

If the series (1) converges, but the series of absolute values of its terms

(3)

does not, then the series is said to be conditionally convergent or  semi-convergent. If the series (3) is convergent then (1) is said to be absolutely convergent.

Cauchy in his Analyse algébrique [1] , p.142 proved:

Theorem: If a series (1) is absolutely convergent then it is convergent.

If (1) is a absolutely convergent series, and   with , then .

References

[1]	Cauchy, A. L. (1821). Cours d'Analyse de l'École Royale Polytechnique: Première Partie: Analyse Algébrique . Paris: Chez Debure frères.

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