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 $
