The set of Mertens (by Franz Mertens ) is a mathematical theorem from the analysis , a statement about convergence of a Cauchy-product of two rows supplies.
Without restriction, let the series be absolutely convergent. To show now is that the partial sum to converge.
In the following be and .
Cauchy product
can be written as
can be written as
The difference formation 1. - 2. gives
It converges to zero and the last row can be split with
It applies
for the last term is a product of a null sequence with a bounded sequence. Since the null sequence has to be restricted, there is a with . thats why
according to the Cauchy criterion . So what follows immediately applies .
The Cauchy product under conditional convergence
If both output series only converge to a limited extent, then the Cauchy product does not have to converge, as the example shows: The Cauchy product of the series with does not converge, see Cauchy product formula # A divergent series .
Hardy showed, however, that the Cauchy product also converges for two series which are only partially convergent if the sequences and are limited. For the known not absolutely convergent output series
with value the Cauchy product is thus convergent with value .
Individual evidence
^ Konrad Königsberger: Analysis 1 - 5th edition. Springer-Verlag Berlin Heidelberg New York, ISBN 3-540-41282-4 ; P. 74 (end of section 6.3)
^ The Multiplication of Conditionally Convergent Series, GH Hardy, Proceedings of the London Mathematical Society, Volume s2-6, Issue 1, 1908, Pages 410-423, https://doi.org/10.1112/plms/s2-6.1.410