Leibniz criterion

The Leibniz criterion is a convergence criterion in the mathematical branch of analysis . With this criterion the convergence of an infinite series can be shown. It is named after the universal scholar Gottfried Wilhelm Leibniz , who published the criterion in 1682.

Statement of the criterion

Partial sum of an alternating series

Let be a monotonically falling , real zero sequence , then the alternating series converges ${\displaystyle (a_{n})_{n\in \mathbb {N} _{0}}}$

${\displaystyle s=\sum _{n=0}^{\infty }(-1)^{n}a_{n}\,.}$

However , the criterion makes no statement about the limit value of the series.

The criterion also applies to monotonically growing zero sequences.

Examples

With the Leibniz criterion, for example, the convergence of the alternating harmonic series and the Leibniz series can be shown.

Alternating harmonic series

The alternating harmonic series

${\displaystyle 1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}\mp \cdots =\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=\ln 2}$

converges according to the Leibniz criterion. However, it does not absolutely converge .

Leibniz series

→ Main article : Leibniz series

${\displaystyle 1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+{\frac {1}{9}}\mp \cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}={\frac {\pi }{4}}}$.

Counterexample

This counterexample shows that it is not enough if there is only one null sequence. The monotony is necessary for this criterion. Considering the non-monotonic zero sequence ${\displaystyle (a_{n})}$

${\displaystyle a_{n}={\begin{cases}0&\mathrm {falls} \ n\ \mathrm {gerade} ,\\{\frac {2}{n+1}}&\mathrm {falls} \ n\ \mathrm {ungerade} .\end{cases}}}$

The alternating series with these coefficients has the negative harmonic series as odd members , which diverges. Hence the entire series is also divergent. ${\displaystyle s}$ ${\displaystyle s}$

Estimation of the limit value

The Leibniz criterion provides an estimate for the limit value, because with such alternating series the limit value is always between two successive partial sums. Be

${\displaystyle s_{N}=\sum _{n=0}^{N}(-1)^{n}a_{n}}$

the -th partial sum of the series ${\displaystyle N}$

${\displaystyle s=\sum _{n=0}^{\infty }(-1)^{n}a_{n}}$

with a monotonically decreasing zero sequence . ${\displaystyle (a_{n})_{n\in \mathbb {N} _{0}}}$

Then applies to all : ${\displaystyle k\in \mathbb {N} _{0}}$

${\displaystyle s_{2k+1}\leq s\leq s_{2k}}$.

There is also an error estimate, i.e. an estimate of the remaining term of the sum after summands: ${\displaystyle N}$

${\displaystyle |s-s_{N}|=\left|\sum _{n=N+1}^{\infty }(-1)^{n}a_{n}\right|\leq a_{N+1}.}$

proof

We consider the subsequence of the sequence of partial sums. Since the sequence is monotonically decreasing, the following applies ${\displaystyle (s_{0},s_{2},s_{4},\dots )=(s_{2k})_{k\in \mathbb {N} _{0}}}$ ${\displaystyle (a_{k})_{k\in \mathbb {N} _{0}}}$

${\displaystyle s_{2k+2}=s_{2k}-a_{2k+1}+a_{2k+2}\leq s_{2k},\quad k\in \mathbb {N} _{0}}$.

That is, the sequence is also monotonically decreasing. It is also limited downwards because ${\displaystyle (s_{2k})_{k\in \mathbb {N} _{0}}}$

${\displaystyle s_{2k}=(a_{0}-a_{1})+(a_{2}-a_{3})+\dots +(a_{2k-2}-a_{2k-1})+a_{2k}\geq a_{2k}\geq 0}$,

after the bracketed expressions are greater than or equal to zero because of the monotony of the sequence . The sequence is therefore not only monotonically decreasing, but also restricted downwards and thus convergent according to the monotony criterion . The sequence is also convergent (similar argument as above, but increasing monotonically) and has the same limit, da ${\displaystyle (a_{k})_{k\in \mathbb {N} }}$ ${\displaystyle (s_{2k})_{k\in \mathbb {N} _{0}}}$ ${\displaystyle (s_{1},s_{3},s_{5},\dots )=(s_{2k+1})_{k\in \mathbb {N} _{0}}}$

${\displaystyle \lim _{k\to \infty }s_{2k+1}=\lim _{k\to \infty }\left(s_{2k}-a_{2k+1}\right)=\lim _{k\to \infty }s_{2k}}$

because of

${\displaystyle \lim _{k\to \infty }a_{2k+1}=0}$

applies.

generalization

The Leibniz criterion is a special case of the more general Dirichlet criterion .

Individual evidence

1. ^ Leibniz criterion . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
2. ↑ See http://mo.mathematik.uni-stuttgart.de/inhalt/erlaeuterung/erlaeuterung286/
3. ^ Proof according to the Handbook of Mathematics. Leipzig 1986, ISBN 3-8166-0015-8 , pp. 408-409. In contrast to this article, the series in the book begins with , so there is a small difference. ${\displaystyle a_{1}}$