# Difference: FamousSeries (1 vs. 2)

#### Revision 22008-11-17 - DickFurnas

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 META TOPICPARENT name="InfiniteSeriesSynopsis"

# Famous Series

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## Geometric Series

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• if converges with sum
• if diverges
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• for diverges
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## Harmonic Series

• diverges
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• 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.
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• converges
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where
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where

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• converges
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• converges
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## ln (1+x)

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• you can arrive at this relation by integrating a Geometric Series in -t term-by-term.
`<--/twistyPlugin twikiMakeVisibleInline-->`
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`<--/twistyPlugin-->`
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• if x=-1 , i.e. ln(1+(-1)) = ln(0) , this is the negative of the Harmonic Series which diverges toward -&infty;

### arctan (x)

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• if x=-1 , i.e. ln(1+(-1)) = ln(0) , this is the negative of the Harmonic Series which diverges toward -∞

## arctan (x)

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• you can arrive at this relation by integrating a Geometric Series in -t^2 term-by-term.
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• you can arrive at this relation by integrating a Geometric Series in -t2 term-by-term.

`<--/twistyPlugin twikiMakeVisibleInline-->`
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`<--/twistyPlugin-->`

#### Revision 12008-11-16 - DickFurnas

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 META TOPICPARENT name="InfiniteSeriesSynopsis"

# Famous Series

## Geometric Series

• if converges with sum
• if diverges

• 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.

• converges

where

• converges

• converges

## ln (1+x)

• you can arrive at this relation by integrating a Geometric Series in -t term-by-term.
`<--/twistyPlugin twikiMakeVisibleInline-->`

`<--/twistyPlugin-->`
• if x=-1 , i.e. 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.
`<--/twistyPlugin twikiMakeVisibleInline-->`

`<--/twistyPlugin-->`

-- DickFurnas - 16 Nov 2008

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