Comparison test for real series

Stub grade: A*

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Flesh out, link, then demote. This is needed for functional analysis

Statement

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

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

Then:

• if [ilmath]\sum^\infty_{n\eq 1}b_n[/ilmath] converges, so does [ilmath]\sum^\infty_{n\eq 1}a_n[/ilmath]
• if [ilmath]\sum^\infty_{n\eq 1}a_n[/ilmath] diverges so does [ilmath]\sum^\infty_{n\eq 1}b_n[/ilmath]

Proof

Case 1

• Uses A monotonically increasing sequence bounded above converges, my notes in the picture are VERY messy but get there in the end.

Grade: C

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Routine for first years so unimportant. Forgive my messy notes

• 

  Crappy notes

Case 2

Grade: C

This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.

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Routine, but would be good to do

References

Grade: C

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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.
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See correspondence with David Guichard on 22/11/2016 for where I sourced this