Maclaurin series
================

We are given the two following Maclaurin series that correspond
respectively to the development of functions :math:`\frac{1}{1-x}`
and *arctangent*:

   :math:`\sum_{n=0}^{k} x^{n} = 1 + x + x^{2} + x^{3} + \cdots`

   :math:`\sum_{n=0}^{k}(-1)^{n} \frac{x^{2n+1}}{2n+1} = x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \cdots`

Save all function into file ``maclaurin.py``.

#. Given a real value ``x`` as :math:`-1 \leq x \leq 1` and another
   positive real value ``epsilon`` which is the tolerance, write the
   function :py:func:`series1` to calculate the sum of terms of the
   first given series. The summation must stop when two consecutive
   terms are equal with a tolerance ``epsilon``, in other words, when
   the absolute value of the difference between them is less than
   ``epsilon``. Examples:

   .. literalinclude:: maclaurin1.txt
      :language: python3
      :lines: 3-
             
   .. note::

      You can download the file with tests :download:`maclaurin1.txt <maclaurin1.txt>`

#. Write the function :py:func:`series2` analogous to function
   :py:func:`series1` but using the second given series. Examples:

   .. literalinclude:: maclaurin2.txt
      :language: python3
      :lines: 3-
             
   .. note::

      You can download the file with tests :download:`maclaurin2.txt <maclaurin2.txt>`

.. rubric:: Solution
      
A solution of these functions is provided in the :download:`maclaurin.py
<solutions/maclaurin.py>`