Sinus of the Inverse Index Series
=================================

Let's consider an infinite mathematical succession defined by:

.. math::

    t_i = k \cdot \sqrt{i} \cdot \sin\left(\frac{1}{i^2}\right)


We wish to compute the sum of these terms (ie. computer the series) till its "good enough" without exceeding a given limit of iterations.

Please implement the following Python function in the :mod:`sininv_series` module (:file:`sininv_series.py` file):

    .. function:: sininv_series(k, ntmax, eps):

such that

    **given**

        - ``k``, a :class:`float`, with the coeficient.
        - ``ntmax``, an :class:`int` such that ``ntmax >= 1``, with the maximum number of iterations.
        - :math:`eps`, a :class:`float` with the tolerance.

    **returns** two values which are:

        #. A :class:`float` with the **sum** of the terms the given succession until either the difference between consecutive terms is smaller than the given ``eps`` or the number of terms added to this sum reached ``ntmax``.

        #. An :class:`int` with the **number of terms** added.

.. note::
    Notice that the term that makes the difference smaller than the tolerance is not added.

For example:

.. literalinclude:: sininv_series.test
    :language: pycon
    :start-after: --ini
    :end-before: --fi

Doctests are available at the :download:`sininv_series.test <sininv_series.test>` file.