sum of series

If a series n=1an of real or complex numbers is convergentMathworldPlanetmathPlanetmath and the limit of its partial sums is S, then S is said to be the sum of the series.  This circumstance may be denoted by


or equivalently


The sum of series has the distributive property


with respect to multiplication.  Nevertheless, one must not think that the sum series means an addition of infinitely many numbers — it’s only a question of the limit

limn(a1+a2++an)partial sum.

See also the entry “manipulating convergent series”!

The sum of the series is equal to the sum of a partial sum and the corresponding remainder term.

Title sum of series
Canonical name SumOfSeries
Date of creation 2014-02-15 19:17:15
Last modified on 2014-02-15 19:17:15
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Definition
Classification msc 40-00
Related topic SumFunctionOfSeries
Related topic ManipulatingConvergentSeries
Related topic RemainderTerm
Related topic RealPartSeriesAndImaginaryPartSeries
Related topic LimitOfSequenceAsSumOfSeries
Related topic PlusSign
Defines partial sum