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
Famous quotes containing the words geometric and/or series:
“New York ... is a city of geometric heights, a petrified desert of grids and lattices, an inferno of greenish abstraction under a flat sky, a real Metropolis from which man is absent by his very accumulation.”
—Roland Barthes (19151980)
“The womans world ... is shown as a series of limited spaces, with the woman struggling to get free of them. The struggle is what the film is about; what is struggled against is the limited space itself. Consequently, to make its point, the film has to deny itself and suggest it was the struggle that was wrong, not the space.”
—Jeanine Basinger (b. 1936)