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Infinite Series: A Synopsis

Table of Contents
Infinite Series: A Synopsis Famous Series Geometric Series P-Series Harmonic Series Alternating Harmonic Series Exponential Sin Cos ln (1+x) arctan (x) Useful Inequalities "Eventually" Tips

Famous Series

Geometric Series

if converges with sum
if diverges

P-Series

for diverges

Harmonic Series

diverges

this is a special case of the P-Series for P=1

Alternating Harmonic Series

converges to ln(1+1) = ln(2) using series for ln(1+x) below.

Exponential

converges

where

Sin

converges

Cos

converges

ln (1+x)

you can arrive at this relation by integrating a Geometric Series in -t term-by-term.

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if x=-1 ( ln(1+(-1)) = ln(0) ) this is the negative of the Harmonic Series which diverges toward -&infty;

arctan (x)

you can arrive at this relation by integrating a Geometric Series in -t^2 term-by-term.

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Useful Inequalities

"Eventually"

for any

(!) $smile$ Remember this when using the Test for Divergence

Tips

Often the hardest part of showing convergence or divergence of a series is the indecision: What do I believe it does? After all, you'll have a tough time showing a series converges if it doesn't!
The limits listed in another section can help a lot with the Test for Divergence. Together with inequalities you can often get an idea of what to try to show. If the individual terms of the series "look like" as then the series "looks like" 1/n and you will want to show it diverges, perhaps even setting up a comparison, or limit comparison with 1/n itself.

-- DickFurnas - 21 Oct 2008

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