Geometric Progression - Geometric Series

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

$\begin{align} (1-r) \sum_{k=0}^{n} ar^k & = (1-r)(ar^0 + ar^1+ar^2+ar^3+\cdots+ar^n) \\ & = ar^0 + ar^1+ar^2+ar^3+\cdots+ar^n \\ & {\color{White}{} = ar^0} - ar^1-ar^2-ar^3-\cdots-ar^n - ar^{n+1} \\ & = a - ar^{n+1} \end{align}$

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:

$\frac{d}{dr}\sum_{k=0}^nr^k = \sum_{k=1}^n kr^{k-1}= \frac{1-r^{n+1}}{(1-r)^2}-\frac{(n+1)r^n}{1-r}.$

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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