AD Series
=========

A *mathematical sucsession* :math:`t_1, t_2, ... , t_i, ...` is an infinite sequence of numbers (usually *real*) such that each :math:`t_i` is computed by a formula parameterized to the previous term (or a number of them).
The *Mathematical Series* induced by a given math succession is the succession of the incremental sums of the succession's terms.
Hence, the series will be :math:`s_1, s_2, ... , s_n, ...` such that

    :math:`s_1 = t_1`

    :math:`s_2 = t_1 + t_2`

    :math:`s_3 = t_1 + t_2 + t_3`

    ...


Let's consider the recursive definition of a mathematical succession parameterized by the real number constants :math:`k`, :math:`a`, and the succession index :math:`i` such that:

    :math:`t_{1} = a`

    :math:`t_{i} =  {-1}^i * \pi * \frac {k + \sqrt{|t_{i-1}|}} {2*|t_{i-1}|}`

Implement the following two Python functions in the module :mod:`ad_series` (file :file:`ad_series.py`).

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The first function is:

    .. py:function:: ad_term(k, x, i)

such that

    **given**

    - ``k``, :class:`float`, is the constant of the above succession
    - ``x``, :class:`float`, is the value of the :math:`x_{i-1}` term of the above succession
    - ``i``, :class:`int`, :math:`i > 1`, the index of the term to be calculated

    **returns** a :class:`float` with the value of the :math:`x_i` term of the succession calculated according to the above definition.

For example:

   .. literalinclude:: ad_term-test.txt
      :language: python3
      :start-after: --ini
      :end-before: --fi

They are available at the :download:`ad_term-test.txt <ad_term-test.txt>` file.

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The second function is:

    ``ad_sum5``:math:`(k, a)`

such that

    **given**

        - ``k`` is a :class:`float` with the constant of the succesion
        - ``a`` is a :class:`float` value of the first term of the succession

    **returns** a :class:`float` with `s_5` of the series induced by the succession above (in either words, the addition of the first 5 terms of the succession) rounded to 4 decimals.

    .. note:: Notice that `a` is the first term in the definition and therefore it is part of the 5 terms to be added.

    For example:

    .. literalinclude:: ad_sum5-test.txt
        :language: python3
        :start-after: --ini
        :end-before: --fi

.. warning:: This function implementation **must** call the previous function as many times as needed.

Doctests for validation are available at the :download:`ad_sum5-test.txt` file.