16.3 Geometric Series
=====================

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Consider the series:

.. math::

  \sum_{k=0}^n c^k

for some :math:`c \neq 0` and :math:`c \neq 1`.

We can calculate the sum by simply multiplying the series by :math:`(1-c)`:

.. math::

  (1 + c + c^2 + c^3 + \dots c^n)(1-c) = 1 - c^{n+1}

Dividing both sides by :math:`(1-c)` yields:

.. math::

  \sum_{k=0}^n c^k = \frac{1}{1-c} - \frac{c^{n+1}}{1-c}

What happens as we increase :math:`n \rightarrow \infty`?

* If :math:`c > 1`, then the term on the right will blow up and also increase
  off to infinity. This isn't very interesting, so let's ignore it for now.
* If :math:`0 < c < 1`, then the term on the right will get smaller and
  smaller and eventually become 0.

This second case is most interesting, because it gives us:

.. math::

  \sum_{k=0}^\infty c^k = \frac{1}{1-c} \text{ when } 0 < c < 1

Note that the :math:`\infty` symbol is not a number. In this context, it is
the idea that :math:`k` keeps increasing on and on forever.

As an example, suppose :math:`c = \frac{1}{2}`:

.. math::

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