# sequence determining convergence of series

Theorem. Let ${a}_{1}+{a}_{2}+\mathrm{\dots}$ be any series of real ${a}_{n}$. If the positive numbers ${r}_{1},{r}_{2},\mathrm{\dots}$ are such that

$\underset{n\to \mathrm{\infty}}{lim}{\displaystyle \frac{{a}_{n}}{{r}_{n}}}=L\ne \mathrm{\hspace{0.17em}0},$ | (1) |

then the series converges simultaneously with the series ${r}_{1}+{r}_{2}+\mathrm{\dots}$

*Proof.* In the case that the limit (1) is positive, the supposition implies that there is an integer ${n}_{0}$ such that

$$ | (2) |

Therefore

$$ |

and since the series ${\sum}_{n=1}^{\mathrm{\infty}}0.5L{r}_{n}$ and ${\sum}_{n=1}^{\mathrm{\infty}}1.5L{r}_{n}$ converge simultaneously with the series ${r}_{1}+{r}_{2}+\mathrm{\dots}$, the comparison test^{} guarantees that the same concerns the given series ${a}_{1}+{a}_{2}+\mathrm{\dots}$

The case where (1) is negative, whence we have

$$\underset{n\to \mathrm{\infty}}{lim}\frac{-{a}_{n}}{{r}_{n}}=-L>0,$$ |

may be handled as above.

Note. For the case $L=0$, see the limit comparison test^{}.

Title | sequence determining convergence of series |
---|---|

Canonical name | SequenceDeterminingConvergenceOfSeries |

Date of creation | 2013-03-22 19:06:54 |

Last modified on | 2013-03-22 19:06:54 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 6 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 40A05 |

Related topic | LimitComparisonTest |