Series

Abstract

It follows from the associative law for addition that, if a₁,…,a _N are a finite number of elements, the sum a₁+a₂+… +a _N gives the same element whichever way brackets are inserted, so this expression has an unambiguous meaning. We now want to give a meaning, if possible, to an expression like a₁+a₂+a₃ +…, where (a _n ) is an infinite sequence. A convenient notation for a finite sum a₁ + a₂ + … +a _N is $\sum\limits_{n = 1}^N {{a_n}}$, and we shall denote the infinite series a₁+a₂+a₃ +… by $\sum\limits_{n = 1}^\infty {{a_n}}$ or just $\sum {{a_n}}.$ In some cases it is possible to find a number which can reasonably be called the sum of the series. In other cases, we can write down the series but cannot define its sum in any sensible way.