where

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

arctan (x)

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

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 , 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.

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-- DickFurnas - 16 Nov 2008