Tricks for dealing with geometric series

After reducing a complicated sum for 10 minutes on a test you hit a roadblock: you forgot once again how to simplify a geometric series.

A geometric series is a sum of the form

$ \sum_{k=0}^{n-1} ar^k=a\frac{1-r^n}{1-r} $ ($ r $ not equal to 1)

A very common variant is the sum to infinity

$ \sum_{k=0}^\infty ar^k = \frac{a}{1-r} $, for r < 1.

Both formulas should be memorized and loved.

For example,

$ \sum_{k=0}^\infty (\frac{1}{2})^k = \frac{1}{1-\frac{1}{2}} = 2 $

Tricks for dealing with geometric series include the following:

$ \sum_{k=-\infty}^0 2^k $

A change of variables $ r = -k\, $ gives:

$ \sum_{k=\infty}^0 2^{-r} = \sum_{k=0}^\infty (\frac{1}{2})^{r} = 2 $

If your sum isn't from zero to infinity you can do the following:

$ \sum_{k=1}^\infty (\frac{1}{2})^k = \sum_{k=0}^\infty (\frac{1}{2})^k - (\frac{1}{2})^k|_{k=0} = 2 - 1 = 1 $

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