Introduction to Infinite Series

A.D. that the wider significance of finite and infinite series was realized. The finite series generally do not involve any difficulty in respect of the validity of application of algebraic operations thereto, as compared to the infinite series. The application of algebraic operations to infinite series requires the additional concept of the convergence of series. Violation of condition of convergence of infinite series may cause serious complications. For example, successive division gives

\frac{1}{{x - 1}} = \frac{1}{x} + \frac{1}{{{x^2}}} + \frac{1}{{{x^3}}} + \cdots

Hence, if we put x = \frac{1}{2}, we get

 - 2 = 2 + {2^2} + {2^3} + \cdots

as absurd result. (The condition violated is \left| x \right| > 1).

The theory behind the validity of such expansions is covered under the study of convergence.

The English mathematicians, Brook Taylor (1685—1731) and James Sterling (1692—1770), and the Scotch mathematician Colin Maclaurin (1698—1746) made important contributions to the study of infinite series. But they too did not specially go into the nature of infinite series as such. The question of convergence of infinite series was first subjected to rigorous investigation by the German mathematician Carl Friedrich Gauss (1777—1855). He is remembered as the prince among mathematicians and is ranked with the greatest two stars in the galaxy of mathematics, Archimedes (225 B.C.) and Sir Isaac Newton (1642—1727). Gauss made fundamental contributions of the highest importance to mathematics and science. Almost every field of pure and applied mathematics has been enriched by his genius.

In this tutorial certain properties of the infinite series have been studied. Some writers use the word progression instead of the word series. But here the word series, which is due to the writers of the 17th century and is most commonly used, is preferred.