Criterion of grief

The criterion of grief (after the German mathematician Ernst Kummer ) is a mathematical convergence criterion , that means for deciding whether an infinite series ( absolute ) converges .

The sorrow criterion contains two statements, about convergence and about divergence.

formulation

Be a positive real number sequence. With this the row is formed. This series should be examined for convergence or divergence. ${\displaystyle (c_{k})_{k\in \mathbb {N} }}$ ${\displaystyle S=\sum _{k=1}^{\infty }c_{k}}$

Convergence statement

Is there a positive real number sequence so that from a certain index the expression ${\displaystyle (\alpha _{k})}$ ${\displaystyle \mu }$

${\displaystyle \alpha _{k-1}\,{\frac {c_{k-1}}{c_{k}}}-\alpha _{k}}$

is always greater than or equal to a positive constant , then the series converges . ${\displaystyle \theta >0}$ ${\displaystyle S=\sum _{k=1}^{\infty }c_{k}}$

Divergence statement

Is there a positive real number sequence such that ${\displaystyle (\alpha _{k})}$

• the series of reciprocal terms diverges and ${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{\alpha _{k}}}}$
• from a certain index the expression ${\displaystyle \mu }$

${\displaystyle \alpha _{k-1}\,{\frac {c_{k-1}}{c_{k}}}-\alpha _{k}}$

is always less than or equal to zero,

then the series diverges . ${\displaystyle S=\sum _{k=1}^{\infty }c_{k}}$

proofs

Proof of the convergence statement

The estimate applies to all indices ${\displaystyle k>\mu }$

${\displaystyle 0<\theta \leq \alpha _{k-1}\,{\frac {c_{k-1}}{c_{k}}}-\alpha _{k}}$.

After multiplying with, this results in ${\displaystyle c_{k}\,}$

${\displaystyle \theta c_{k}\leq \alpha _{k-1}c_{k-1}-\alpha _{k}c_{k}}$.

This inequality can now be added up to any large natural number in the manner of a telescope sum. ${\displaystyle k=\mu +1\,}$ ${\displaystyle n>\mu \,}$

${\displaystyle \theta \sum _{k=\mu +1}^{n}c_{k}\leq \sum _{k=\mu +1}^{n}(\alpha _{k-1}c_{k-1}-\alpha _{k}c_{k})=\alpha _{\mu }\,c_{\mu }-\alpha _{n}c_{n}}$

The last term is always less than , this bound does not depend on . So applies to everyone ${\displaystyle \alpha _{\mu }\,c_{\mu }}$ ${\displaystyle n\,}$ ${\displaystyle n>\mu \,}$

${\displaystyle \sum _{k=\mu +1}^{n}c_{k}\leq {\frac {\alpha _{\mu }\,c_{\mu }}{\theta }}}$

Therefore the sequence of partial sums grows monotonically from the index and is limited upwards. According to the (trivial) criterion of monotonic convergence, this converges . ${\displaystyle S_{n}=\sum _{k=1}^{n}c_{k}}$ ${\displaystyle \mu \,}$ ${\displaystyle S=\sum _{k=1}^{\infty }c_{k}}$

Proof of the divergence statement

The estimate applies to all indices ${\displaystyle k>\mu }$

${\displaystyle \alpha _{k-1}\,{\frac {c_{k-1}}{c_{k}}}-\alpha _{k}\leq 0}$and so too . ${\displaystyle \alpha _{k}c_{k}\geq \alpha _{k-1}c_{k-1}}$

The inductive concatenation of these inequalities of up to an arbitrarily large index results ${\displaystyle k=\mu +1\,}$ ${\displaystyle m>\mu }$

${\displaystyle \alpha _{m}c_{m}\geq \alpha _{\mu }c_{\mu }}$,

after further adjustment

${\displaystyle c_{m}\geq {\frac {\alpha _{\mu }}{\alpha _{m}}}c_{\mu }}$.

If this inequality is added up to an arbitrarily large index , it follows ${\displaystyle m=\mu +1}$ ${\displaystyle n}$

${\displaystyle \sum _{m=\mu +1}^{n}c_{m}\geq \alpha _{\mu }c_{\mu }\sum _{m=\mu +1}^{n}{\frac {1}{\alpha _{m}}}}$

Last row diverges according to requirement for . So also diverges according to the minorant criterion . ${\displaystyle n\to \infty }$ ${\displaystyle S=\sum _{m=1}^{\infty }c_{m}}$

Individual evidence

1. ↑ ^{a b} Vladimir Smirnow : Course of higher mathematics , Harri Deutsch Verlag, ISBN 3-8171-1419-2 , pp. 309-310.