# Leibniz’ estimate for alternating series

Theorem (Leibniz 1682).
If ${p}_{1}>{p}_{2}>{p}_{3}>\mathrm{\cdots}$ and $\underset{m\to \mathrm{\infty}}{lim}{p}_{m}=0$, then the alternating series^{}

${p}_{1}-{p}_{2}+{p}_{3}-{p}_{4}+-\mathrm{\dots}$ | (1) |

converges. Its remainder term has the same sign (http://planetmath.org/SignumFunction) as the first omitted $\pm {p}_{m+1}$ and the absolute value^{} less than ${p}_{m+1}$.

Proof. The convergence of (1) is proved here (http://planetmath.org/ProofOfAlternatingSeriesTest). Now denote the sum of the series by $S$ and the partial sums of it by ${S}_{1},{S}_{2},{S}_{3},\mathrm{\dots}$. Suppose that (1) is truncated after a negative $-{p}_{2n}$. Then the remainder term

$${R}_{2n}=S-{S}_{2n}$$ |

may be written in the form

$${R}_{2n}=({p}_{2n+1}-{p}_{2n+2})+({p}_{2n+3}-{p}_{2n+4})+\mathrm{\dots}$$ |

or

$${R}_{2n}={p}_{2n+1}-({p}_{2n+2}-{p}_{2n+3})-({p}_{2n+4}-{p}_{2n+5})-\mathrm{\dots}$$ |

The former shows that ${R}_{2n}$ is positive as the first omitted ${p}_{2n+1}$ and the latter that $$. Similarly one can see the assertions true when the series (1) is truncated after a positive ${p}_{2n-1}$.

*A pictorial proof.*

As seen in this diagram, whenever ${m}^{\prime}>m$, we have
$|{S}_{{m}^{\prime}}-{S}_{m}|\le {p}_{m+1}\to 0$. Thus the partial sums form a Cauchy sequence, and hence converge. The limit lies in the of the spiral, strictly in ${S}_{m}$ and ${S}_{m+1}$ for any $m$. So the remainder after the $m$th must have the same direction as $\pm {p}_{m+1}={S}_{m+1}-{S}_{m}$ and lesser magnitude.

Example 1. The alternating series

$$\frac{1}{\sqrt{2}-1}-\frac{1}{\sqrt{2}+1}+\frac{1}{\sqrt{3}-1}-\frac{1}{\sqrt{3}+1}+\frac{1}{\sqrt{4}-1}-\frac{1}{\sqrt{4}+1}+-\mathrm{\dots}$$ |

does not fulfil the requirements of the theorem and is divergent.

Example 2. The alternating series

$$\frac{1}{\mathrm{ln}2}-\frac{1}{\mathrm{ln}3}+\frac{1}{\mathrm{ln}4}-\frac{1}{\mathrm{ln}5}+-\mathrm{\dots}$$ |

satisfies all conditions of the theorem and is convergent^{}.

Title | Leibniz’ estimate for alternating series |
---|---|

Canonical name | LeibnizEstimateForAlternatingSeries |

Date of creation | 2014-07-22 15:34:38 |

Last modified on | 2014-07-22 15:34:38 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 35 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 40A05 |

Classification | msc 40-00 |

Synonym | Leibniz’ estimate for remainder term |

Related topic | EIsIrrational2 |

Related topic | ConvergingAlternatingSeriesNotSatisfyingAllLeibnizConditions |