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Geometric Series

number line from 0 to 1 ... add ( 1 / 2 ) + ( 1 / 4 )
 

number line from 0 to 1 ... add ( 1 / 2 ) + ( 1 / 4 ) + ( 1 / 8 )


number line from 0 to 1 ... add ( 1 / 2 ) + ( 1 / 4 ) + ( 1 / 8 ) + ( 1 / 16 )

 

See what's happening?  Can you guess what the sum will be?

Let's crunch the formula: a1 = ( 1 / 2 )
,		 r = ( 1 / 2 )

the summation of ( ( 1 / 2 ) * ( 1 / 2 )^( k - 1 ) ) as k goes from 1 to infinity = ( 1 / 2 ) / ( 1 - ( 1 / 2 ) ) = ( 1 / 2 ) / ( 1 / 2 ) = 1

Do you believe it?  Repeat after me:  Duuuuude.

By the way, when | r | < 1 and we CAN find the sum, the series
is called "convergent."