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Homesum of series

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# 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.

Defines:

partial sum

Related:

SumFunctionOfSeries, ManipulatingConvergentSeries, RemainderTerm, RealPartSeriesAndImaginaryPartSeries, LimitOfSequenceAsSumOfSeries, PlusSign

Major Section:

Reference

Type of Math Object:

Definition

Parent:

Groups audience:

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sometthing... by drini ✘