# 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.

• 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.

-- DickFurnas - 16 Nov 2008

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Topic revision: r1 - 2008-11-16 - 22:05:11 - DickFurnas

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