# Comparison test for real series

From Maths

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

## Contents

## 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:

- [ilmath]\forall n\in\mathbb{N}[a_n\ge 0\wedge b_n\ge 0][/ilmath] - neither sequence is non-negative, and
- [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.

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### Case 2

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Routine, but would be good to do

## References

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

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