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Geometric Series

OK, now for the mind-blowing infinite thing!

Now, we want to find this sum:

the summation of ( 3( 1 / 2 )^( k - 1 ) ) as k goes from 1 to infinity ... the infinity means that there are an infinite number of numbers!
 

And...  we WILL get a finite number for the answer!

Here's the formula...

           For this to work, the ratio has got to be greater than -1
           and less than 1...

number line showing the ratio is between -1 and 1

(The ratio must be between -1 and 1.)

The math way to say this is| r | < 1.

So...

If| r | < 1, then

S infinity = the summation of ( a1 * r^( k -1 ) ) as k goes from 1 to infinity = a1 / ( 1 - r )

That's it!