A geometric series is the sum of the numbers in a geometric progression:
We can find a simpler formula for this sum by multiplying both sides of the above equation by 1 − r, and we'll see that
since all the other terms cancel. If r ≠ 1, we can rearrange the above to get the convenient formula for a geometric series:
If one were to begin the sum not from k=0, but from a higher term, say m, then
Differentiating this formula with respect to r allows us to arrive at formulae for sums of the form
For example:
For a geometric series containing only even powers of r multiply by 1 − r2 :
Then
Equivalently, take r2 as the common ratio and use the standard formulation.
For a series with only odd powers of r
and
Read more about this topic: Geometric Progression
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