Comparison test for real series/Statement

Grade: D

This page requires references, it is on a to-do list for being expanded with them.

Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:

Routine, but a reference would be good

Statement

Suppose $(a_n)_{n\in\mathbb{N} }$ and $(b_n)_{n\in\mathbb{N} }$ are real sequences and that we have:

1. $\forall n\in\mathbb{N}[a_n\ge 0\wedge b_n\ge 0]$ - neither sequence is non-negative, and
2. $\exists K\in\mathbb{N}\forall n\in\mathbb{N}[n>K\implies b_n\ge a_n]$ - i.e. that eventually $b_n\ge a_n$.

Then:

• if $\sum^\infty_{n\eq 1}b_n$ converges, so does $\sum^\infty_{n\eq 1}a_n$
• if $\sum^\infty_{n\eq 1}a_n$ diverges so does $\sum^\infty_{n\eq 1}b_n$