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Geometric Series

OK, this is going to blow your mind!  In this section, I'm going to add up an infinite number of numbers -- all positive -- and get a FINITE answer!

Think about it...

If you add these guys... forever...

1 + 2 + 4 + 8 + 16 + ...

The answer would be infinite -- right?

Well, I'm going to get a number.  Just wait.  We'll get to it in awhile.

First, let's check out the sigma notation for geometric sequences:

the summation of ( a1 * r^( k - 1 ) ) as k goes from 1 to infinity ... Remember that this comes from the nth term: an = a1 * r^( n - 1 )

Here's an example:

the summation of ( 3( -2 )^( k - 1 ) ) as k goes from 1 to infinity = 3 - 6 + 12 - 24 + ...

 

TRY IT:

Find the sigma notation for

4 + 2 + 1 + ( 1 / 2 ) + ( 1 / 4 ) + ...

* hint: Find r first, then get the nth term formula.