sum of series

If a series ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n$ 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

$$\sum _{n=1}^{\mathrm{\infty }}{a}_n=S$$

or equivalently

$$a_1+a_2+a_3+\mathrm{\dots }=S.$$

The sum of series has the distributive property

$$c(a_1+a_2+a_3+\mathrm{\dots })=c{a}_1+c{a}_2+c{a}_3+\mathrm{\dots }$$

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

$$\underset{n\to \mathrm{\infty }}{lim}\underset{\text{partial sum}}{\underbrace{(a_1+a_2+\mathrm{\dots }+a_n)}}.$$

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