series

Given a sequence of numbers (real or complex) $\{a_n\}$ we define a sequence of partialsums $\{S_N\}$, where $S_N={\sum }_{n=1}^N{a}_n$. This sequence is called the series with terms $a_n$. We define the sum of the series${\sum }_{n=1}^{\mathrm{\infty }}{a}_n$ to be the limit of these partial sums. More precisely

$$\sum _{n=1}^{\mathrm{\infty }}{a}_n=\underset{N\to \mathrm{\infty }}{lim}{S}_n=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=1}^N{a}_n.$$

In a context where this distinction does not matter much (this is usually the case) one identifies a series with its sum, if the latter exists.

Traditionally, as above, series are infinite sums of real numbers. However,the formal constraints on the terms $\{a_n\}$ are much less strict. We needonly be able to add the terms and take the limit of partial sums. So in fullgenerality the terms could be complex numbers or even elements of certain rings,fields, and vector spaces.