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[parent] sum of series (Definition)

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 the sum of the series. This circumstance may be denoted by

$\displaystyle \sum_{n = 1}^\infty a_n = S$
or equivalently
$\displaystyle a_1+a_2+a_3+\cdots = S.$
Nevertheless, one should not think that this means an addition of infinitely many numbers -- it's only a question of the limit
$\displaystyle \lim_{n\to\infty}\underbrace{(a_1+a_2+\cdots+a_n)}_{\textrm{partial sum}}.$

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



"sum of series" is owned by pahio.
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See Also: sum function of series, manipulating convergent series, remainder term, real part series and imaginary part series, limit of sequence as sum of series, plus sign

Also defines:  partial sum

This object's parent.

Attachments:
uncountable sums of positive numbers (Definition) by rspuzio
limit of sequence as sum of series (Theorem) by pahio
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Cross-references: remainder term, numbers, addition, sum, limit, convergent, complex numbers, real, series
There are 42 references to this entry.

This is version 9 of sum of series, born on 2004-11-18, modified 2007-12-02.
Object id is 6493, canonical name is SumOfSeries.
Accessed 12253 times total.

Classification:
AMS MSC: 40-00 (Sequences, series, summability :: General reference works )
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