# sum of series

If a series $\sum_{n=1}^{\infty}a_{n}$ of real or complex numbers is convergent 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

 $\sum_{n=1}^{\infty}a_{n}\;=\;S$

or equivalently

 $a_{1}+a_{2}+a_{3}+\ldots\;=\;S.$

The sum of series has the distributive property

 $c\,(a_{1}+a_{2}+a_{3}+\ldots)\;=\;ca_{1}+ca_{2}+ca_{3}+\ldots$

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

 $\lim_{n\to\infty}\underbrace{(a_{1}+a_{2}+\ldots+a_{n})}_{\textrm{partial sum}}.$

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