def is_convergent(self):
        r"""Checks for the convergence of a Sum.

        We divide the study of convergence of infinite sums and products in
        two parts.

        First Part:
        One part is the question whether all the terms are well defined, i.e.,
        they are finite in a sum and also non-zero in a product. Zero
        is the analogy of (minus) infinity in products as :math:`e^{-\infty} = 0`.

        Second Part:
        The second part is the question of convergence after infinities,
        and zeros in products, have been omitted assuming that their number
        is finite. This means that we only consider the tail of the sum or
        product, starting from some point after which all terms are well
        defined.

        For example, in a sum of the form:

        .. math::

            \sum_{1 \leq i < \infty} \frac{1}{n^2 + an + b}

        where a and b are numbers. The routine will return true, even if there
        are infinities in the term sequence (at most two). An analogous
        product would be:

        .. math::

            \prod_{1 \leq i < \infty} e^{\frac{1}{n^2 + an + b}}

        This is how convergence is interpreted. It is concerned with what
        happens at the limit. Finding the bad terms is another independent
        matter.

        Note: It is responsibility of user to see that the sum or product
        is well defined.

        There are various tests employed to check the convergence like
        divergence test, root test, integral test, alternating series test,
        comparison tests, Dirichlet tests. It returns true if Sum is convergent
        and false if divergent and NotImplementedError if it can not be checked.

        References
        ==========

        .. [1] https://en.wikipedia.org/wiki/Convergence_tests

        Examples
        ========

        >>> from sympy import factorial, S, Sum, Symbol, oo
        >>> n = Symbol('n', integer=True)
        >>> Sum(n/(n - 1), (n, 4, 7)).is_convergent()
        True
        >>> Sum(n/(2*n + 1), (n, 1, oo)).is_convergent()
        False
        >>> Sum(factorial(n)/5**n, (n, 1, oo)).is_convergent()
        False
        >>> Sum(1/n**(S(6)/5), (n, 1, oo)).is_convergent()
        True

        See Also
        ========

        Sum.is_absolutely_convergent()

        Product.is_convergent()
        """
        from sympy import Interval, Integral, Limit, log, symbols, Ge, Gt, simplify
        p, q = symbols('p q', cls=Wild)

        sym = self.limits[0][0]
        lower_limit = self.limits[0][1]
        upper_limit = self.limits[0][2]
        sequence_term = self.function

        if len(sequence_term.free_symbols) > 1:
            raise NotImplementedError("convergence checking for more that one symbol "
                                        "containing series is not handled")

        if lower_limit.is_finite and upper_limit.is_finite:
            return S.true

        # transform sym -> -sym and swap the upper_limit = S.Infinity
        # and lower_limit = - upper_limit
        if lower_limit is S.NegativeInfinity:
            if upper_limit is S.Infinity:
                return Sum(sequence_term, (sym, 0, S.Infinity)).is_convergent() and \
                        Sum(sequence_term, (sym, S.NegativeInfinity, 0)).is_convergent()
            sequence_term = simplify(sequence_term.xreplace({sym: -sym}))
            lower_limit = -upper_limit
            upper_limit = S.Infinity

        interval = Interval(lower_limit, upper_limit)

        # Piecewise function handle
        if sequence_term.is_Piecewise:
            for func_cond in sequence_term.args:
                if func_cond[1].func is Ge or func_cond[1].func is Gt or func_cond[1] == True:
                    return Sum(func_cond[0], (sym, lower_limit, upper_limit)).is_convergent()
            return S.true

        ###  -------- Divergence test ----------- ###
        try:
            lim_val = limit(sequence_term, sym, upper_limit)
            if lim_val.is_number and lim_val is not S.Zero:
                return S.false
        except NotImplementedError:
            pass

        try:
            lim_val_abs = limit(abs(sequence_term), sym, upper_limit)
            if lim_val_abs.is_number and lim_val_abs is not S.Zero:
                return S.false
        except NotImplementedError:
            pass

        order = O(sequence_term, (sym, S.Infinity))

        ### --------- p-series test (1/n**p) ---------- ###
        p1_series_test = order.expr.match(sym**p)
        if p1_series_test is not None:
            if p1_series_test[p] < -1:
                return S.true
            if p1_series_test[p] > -1:
                return S.false

        p2_series_test = order.expr.match((1/sym)**p)
        if p2_series_test is not None:
            if p2_series_test[p] > 1:
                return S.true
            if p2_series_test[p] < 1:
                return S.false

        ### ----------- root test ---------------- ###
        lim = Limit(abs(sequence_term)**(1/sym), sym, S.Infinity)
        lim_evaluated = lim.doit()
        if lim_evaluated.is_number:
            if lim_evaluated < 1:
                return S.true
            if lim_evaluated > 1:
                return S.false

        ### ------------- alternating series test ----------- ###
        dict_val = sequence_term.match((-1)**(sym + p)*q)
        if not dict_val[p].has(sym) and is_decreasing(dict_val[q], interval):
            return S.true

        ### ------------- comparison test ------------- ###
        # (1/log(n)**p) comparison
        log_test = order.expr.match(1/(log(sym)**p))
        if log_test is not None:
            return S.false

        # (1/(n*log(n)**p)) comparison
        log_n_test = order.expr.match(1/(sym*(log(sym))**p))
        if log_n_test is not None:
            if log_n_test[p] > 1:
                return S.true
            return S.false

        # (1/(n*log(n)*log(log(n))*p)) comparison
        log_log_n_test = order.expr.match(1/(sym*(log(sym)*log(log(sym))**p)))
        if log_log_n_test is not None:
            if log_log_n_test[p] > 1:
                return S.true
            return S.false

        # (1/(n**p*log(n))) comparison
        n_log_test = order.expr.match(1/(sym**p*log(sym)))
        if n_log_test is not None:
            if n_log_test[p] > 1:
                return S.true
            return S.false

        ### ------------- integral test -------------- ###
        if is_decreasing(sequence_term, interval):
            integral_val = Integral(sequence_term, (sym, lower_limit, upper_limit))
            try:
                integral_val_evaluated = integral_val.doit()
                if integral_val_evaluated.is_number:
                    return S(integral_val_evaluated.is_finite)
            except NotImplementedError:
                pass

        ### -------------- Dirichlet tests -------------- ###
        if order.expr.is_Mul:
            a_n, b_n = order.expr.args[0], order.expr.args[1]
            m = Dummy('m', integer=True)

            def _dirichlet_test(g_n):
                try:
                    ing_val = limit(Sum(g_n, (sym, interval.inf, m)).doit(), m, S.Infinity)
                    if ing_val.is_finite:
                        return S.true
                except NotImplementedError:
                    pass

            if is_decreasing(a_n, interval):
                dirich1 = _dirichlet_test(b_n)
                if dirich1 is not None:
                    return dirich1

            if is_decreasing(b_n, interval):
                dirich2 = _dirichlet_test(a_n)
                if dirich2 is not None:
                    return dirich2

        raise NotImplementedError("The algorithm to find the Sum convergence of %s "
                                    "is not yet implemented" % (sequence_term))
dataset/ETHPy150Open sympy/sympy/sympy/concrete/summations.py/Sum.is_convergent/original