# convergent series where not only${a}_{n}$ but also $n{a}_{n}$ tends to 0

Proposition. If the terms (http://planetmath.org/Series) ${a}_{n}$ of the convergent series^{}

$${a}_{1}+{a}_{2}+\mathrm{\dots}$$ |

are positive and form a monotonically decreasing sequence^{}, then

$\underset{n\to \mathrm{\infty}}{lim}n{a}_{n}=\mathrm{\hspace{0.33em}0}.$ | (1) |

*Proof.* Let $\epsilon $ be any positive number. By the Cauchy criterion for convergence and the positivity of the terms, there is a positive integer $m$ such that

$$ |

Since the sequence ${a}_{1},{a}_{2},\mathrm{\dots}$ is decreasing, this implies

$$ | (2) |

Choosing here especially $p:=m$, we get

$$ |

whence again due to the decrease,

$$ | (3) |

Adding the inequalities^{} (2) and (3) with the common values $p=m,m+1,\mathrm{\dots}$ then yields

$$ |

This may be written also in the form

$$ |

which means that ${lim}_{n\to \mathrm{\infty}}n{a}_{n}=\mathrm{\hspace{0.33em}0}$.

Remark. The assumption of monotonicity in the Proposition is essential. I.e., without it, one cannot gererally get the limit result (1). A counterexample would be the series ${a}_{1}+{a}_{2}+\mathrm{\dots}$ where
${a}_{n}:=\frac{1}{n}$ for any perfect square^{} $n$ but 0 for other values of $n$. Then this series is convergent^{} (cf. the over-harmonic series), but $n{a}_{n}=1$ for each perfect square $n$; so $n{a}_{n}\to \u03380$ as $n\to \mathrm{\infty}$.

Title | convergent series where not only${a}_{n}$ but also $n{a}_{n}$ tends to 0 |
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Canonical name | ConvergentSeriesWhereNotOnlyanButAlsoNanTendsTo0 |

Date of creation | 2013-03-22 19:03:29 |

Last modified on | 2013-03-22 19:03:29 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 14 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 40A05 |

Synonym | Olivier’s theorem |

Related topic | NecessaryConditionOfConvergence |

Related topic | AGeneralisationOfOlivierCriterion |