Given a sequence of numbers (real or complex) {an} we define a sequence of partial sums {SN}, where SN=n=1Nan. This sequence is called the series with terms an. We define the sum of the series n=1an to be the limit of these partial sums. More precisely


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 {an} are much less strict. We need only be able to add the terms and take the limit of partial sums. So in full generality the terms could be complex numbersMathworldPlanetmathPlanetmath or even elements of certain rings, fields, and vector spaces.

Title series
Canonical name Series
Date of creation 2013-03-22 12:41:35
Last modified on 2013-03-22 12:41:35
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 7
Author mathwizard (128)
Entry type Definition
Classification msc 40-00
Related topic AbsoluteConvergence
Related topic HarmonicNumber
Related topic CompleteUltrametricField
Related topic Summation
Related topic PrimeHarmonicSeries