## Geometric Series

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A geometric series is one in which there is a constant ratio between each element and the one preceding it. Here is one such series.

7+14+28+56+112

The ratio here is 2. Let's try to find the sum of geometric series, in general (S is the sum, a is the first element, r is the ratio, n is the number of elements):

S=a+ar+ar2+ar3+...+arn-1
Sr=ar+ar2+ar3+...+arn-1+arn
S-Sr=a-arn [subtracting the second line from the first]
S(1-r)=a(1-rn)
S=a(1-rn)/(1-r)

That last line is a handy formula. Let's use it to determine the sum of our sample series, above. a=7, r=2, n=5. Then S=7(1-32)/(1-2)=217. Checking our work, we see that 217 is right.

A geometric series is often infinite. In other words, it has infinitely many elements:

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

It should be obvious that such a series diverges (goes to infinity) when r>=1. When -1<r<1 (r is between -1 and 1), the sum approaches a real number. We use limits to find the handy formula for such a sum. Let me explain it in words instead. We take our formula for n terms (above), S=a(1-rn)/(1-r). With -1<r<1, rn gets closer and closer to 0, as n gets large. None of the rest of our formula is affected by our large n. So, we can stick 0 in for rn, and we get:

S=a/(1-r)

In other words, every geometric series, in which the terms get closer and closer to zero, converges to a real number. And we can calculate that number with the previous equation.

And our familiar infinite series (a=1 and r=1/2):

1+1/2+1/4+1/8+1/16...=2