# 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

$$\underset{n\to \mathrm{\infty}}{lim}\underset{\text{partial sum}}{\underset{\u23df}{({a}_{1}+{a}_{2}+\mathrm{\dots}+{a}_{n})}}.$$ |

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 |