Series

Abstract

• Let (a _k) be a sequence of real numbers. We use the notation $s_n \, = \,\sum\limits_{k = 0}^n {a_k }$ to denote the nth partial sum of the infinite series $s_\infty \, = \,\sum\limits_{k = 0}^\infty {a_k }$. If the sequence of partial sums (s _n) converges to a real number s, we say that the series $\sum\limits_k {a_k }$ is convergent and we write $s\, = \sum\limits_{k = 0}^\infty {a_k } \,$. A series that is not convergent is called divergent.

That fondness for science, … that affability and condescension which God shows to the learned, that promptitude with which he protects and supports them in the elucidation of obscurities and in the removal of difficulties, has encouraged me to compose a short work on calculating by al-jabr and almuqabala, confining it to what is easiest and most useful in arithmetic. [al-jabr means “restoring,” referring to the process of moving a subtracted quantity to the other side of an equation; al-muqabala is “comparing” and refers to subtracting equal quantities from both sides of an equation.]

Musa Al-Khwarizmi (about 790–about 840)