(New page: The Geometric Series formulas below still hold for |alpha|'s containing complex exponentials. For k from 0 to n, where |alpha| does not equal 1: <math> \sum_{k=0}^{n} \alpha^k = \frac{1...) |
m (Copy edit: Math formatting) |
||
| (One intermediate revision by one other user not shown) | |||
| Line 1: | Line 1: | ||
| − | The Geometric Series formulas below still hold for | + | The Geometric Series formulas below still hold for <math>\alpha</math>'s containing complex exponentials. |
| − | For k from 0 to n, where | + | For k from 0 to n, where <math>\alpha</math> does not equal 1: |
<math> \sum_{k=0}^{n} \alpha^k = \frac{1-\alpha^{n+1}}{1-\alpha} </math> | <math> \sum_{k=0}^{n} \alpha^k = \frac{1-\alpha^{n+1}}{1-\alpha} </math> | ||
| Line 9: | Line 9: | ||
| − | For k from 0 to infinity, where | + | For k from 0 to infinity, where <math>alpha</math> is less than 1: |
<math>\sum_{k=0}^\infty \alpha^k = \frac{1}{1-\alpha}</math> | <math>\sum_{k=0}^\infty \alpha^k = \frac{1}{1-\alpha}</math> | ||
| Line 23: | Line 23: | ||
in the above Geometric Series formula. | in the above Geometric Series formula. | ||
| + | |||
| + | [[Homework Problem 5.31_OldKiwi]] | ||
Latest revision as of 18:06, 4 April 2008
The Geometric Series formulas below still hold for $ \alpha $'s containing complex exponentials.
For k from 0 to n, where $ \alpha $ does not equal 1:
$ \sum_{k=0}^{n} \alpha^k = \frac{1-\alpha^{n+1}}{1-\alpha} $
(else, = n + 1)
For k from 0 to infinity, where $ alpha $ is less than 1:
$ \sum_{k=0}^\infty \alpha^k = \frac{1}{1-\alpha} $
(else it diverges)
Example: We want to evaluate the following:
$ \sum_{k=0}^\infty (\frac{1}{2})^k e^{-j \omega k}= \sum_{k=0}^\infty (\frac{1}{2}e^{-j\omega})^k = \frac{1}{1-\frac{1}{2}e^{-j\omega}} $
In this case $ \alpha=\frac{1}{2}e^{-j\omega} $
in the above Geometric Series formula.
