def spence_series_diff(b, a):
    """ Returns the Taylor series for Li\ :sub:`2`\ (b) - Li\ :sub:`2`\ (a).

    Li2 is the polylogarithm function defined by
    
    .. math::
       \\text{Li}_2(z) \\equiv \\sum_{k=1}^\\infty \\frac{z^k}{k^2} \\,.

    Parameters
    ----------
    a : ndarray
        Input for Li\ :sub:`2`\ (a).
    b : ndarray
        Input for Li\ :sub:`2`\ (b).

    Returns
    -------
    ndarray
        The Taylor series Li\ :sub:`2`\ (b) - Li\ :sub:`2`\ (a), up to the 11th order term.

    """

    return(
        (b - a) + (b**2 - a**2)/2**2 + (b**3 - a**3)/3**2
        + (b**4 - a**4)/4**2 + (b**5 - a**5)/5**2
        + (b**6 - a**6)/6**2 + (b**7 - a**7)/7**2
        + (b**8 - a**8)/8**2 + (b**9 - a**9)/9**2
        + (b**10 - a**10)/10**2 + (b**11 - a**11)/11**1
    )