View source for Comparison test for real series/Statement

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{{Requires references|grade=D|msg=Routine, but a reference would be good}}
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==Statement==
</noinclude>Suppose {{M|(a_n)_{n\in\mathbb{N} } }} and {{M|(b_n)_{n\in\mathbb{N} } }} are [[real sequences]] and that we have:
# {{M|\forall n\in\mathbb{N}[a_n\ge 0\wedge b_n\ge 0]}} - neither sequence is non-negative, and
# {{M|\exists K\in\mathbb{N}\forall n\in\mathbb{N}[n>K\implies a_n\le b_n}} - i.e. that {{link|eventually|sequence}} {{M|a_n\le b_n}}.
Then:
* if {{M|\sum^\infty_{n\eq 1}b_n}} {{link|converges|sequence}}, so does {{M|\sum^\infty_{n\eq 1}a_n}}
* if {{M|\sum^\infty_{n\eq 1}a_n}} {{link|diverges|series}} so does {{M|\sum^\infty_{n\eq 1}b_n}}<noinclude>
==References==
<references/>
{{Theorem Of|Functional Analysis|Analysis|Metric Space|Real Analysis}}
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