Introduction to Infinite Series

Finite series generally do not involve any difficulty with respect to the validity of the application of algebraic operations 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 the conditions 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 an absurd result. (The condition violates $$\left| x \right| > 1$$).

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 Scotish 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 the convergence of infinite series was first subjected to rigorous investigation by the German mathematician Carl Friedrich Gauss (1777—1855). He is remembered as a significant mathematician 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 were 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.