Difference: FamousSeries (1 vs. 3)

Revision 32008-11-18 - Main.DickFurnas

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META TOPICPARENT name="InfiniteSeriesSynopsis"

Famous Series

Revision 22008-11-17 - Main.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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P-Series

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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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Exponential

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  • converges
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where
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where

 

Sin

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  • converges
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Cos

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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.
More... Close
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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.
  More... Close
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Revision 12008-11-16 - Main.DickFurnas

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META TOPICPARENT name="InfiniteSeriesSynopsis"

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.
More... Close
<--/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.
More... Close
<--/twistyPlugin twikiMakeVisibleInline-->

<--/twistyPlugin-->

-- DickFurnas - 16 Nov 2008

 
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