Wrong result in the convergence of infinite sums with factorials

[maurogaravello]

>>> n = Symbol('n', integer = True)
>>> expr = factorial(n) / factorial(n+2)
>>> s = Sum(expr, (n, 1, oo))
>>> print(s.is_convergent())
False

The correct answer is True

[asmeurer]

Sum(simplify(expr), (n, 1, oo)).is_convergent() gives the right answer (for some reason simplify(s) doesn't work).

[maurogaravello]

The different behaviour originates in the code line

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

If you use simplify, then order is n**(-2). In the other case is 1.
Both results are correct, even if the latter is not sharp.
I think that the code lines

            if p1_series_test[p] >= -1:
                return S.false

are not logically correct, since, in case order does not return a sharp result, these lines can produce a wrong result.

[AnimeshSinha1309]

Can I take up this issue and try to fix it?

[maurogaravello]

@AnimeshSinha1309 Sure.

[AnimeshSinha1309]

Is it not just possible to add the line

sequence_term = simplify(self.function)

Simplifying the terms first should always make it easier to test the convergence of any series summation.

[czgdp1807]

Current status,

>>> n = Symbol('n', integer=True)
>>> expr = factorial(n)/factorial(n+2)
>>> s = Sum(expr, (n, 1, oo))
>>> print(s.is_convergent())
Traceback (most recent call last):
  File "/usr/lib/python3.6/code.py", line 91, in runcode
    exec(code, self.locals)
  File "<console>", line 1, in <module>
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/concrete/summations.py", line 616, in is_convergent
    "is not yet implemented" % (sequence_term))
NotImplementedError: The algorithm to find the Sum convergence of factorial(n)/factorial(n + 2) is not yet implemented

It looks like the issue is outdated. May be we can close this.

[czgdp1807]

>>> n = Symbol('n', integer=True)
>>> expr = factorial(n)/factorial(n+2)
>>> s = Sum(expr, (n, 1, oo))
>>> print(s.is_convergent())
^CTraceback (most recent call last):
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/assumptions.py", line 454, in getit
    return self._assumptions[fact]
KeyError: 'zero'

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/assumptions.py", line 454, in getit
    return self._assumptions[fact]
KeyError: 'extended_positive'

During handling of the above exception, another exception occurred:

Traceback (most recent call last):
  File "/usr/lib/python3.6/code.py", line 91, in runcode
    exec(code, self.locals)
  File "<console>", line 1, in <module>
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/concrete/summations.py", line 458, in is_convergent
    lim_val = limit_seq(sequence_term, sym)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/series/limitseq.py", line 232, in limit_seq
    L1 = _limit_seq(expr.xreplace({n: n_}), n_, trials)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/series/limitseq.py", line 119, in _limit_seq
    result = _limit_inf(expr, n)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/series/limitseq.py", line 109, in _limit_inf
    return Limit(expr, n, S.Infinity).doit(deep=False)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/series/limits.py", line 260, in doit
    r = gruntz(e, z, z0, dir)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/series/gruntz.py", line 669, in gruntz
    r = limitinf(e0, z)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/cache.py", line 94, in wrapper
    retval = cfunc(*args, **kwargs)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/series/gruntz.py", line 433, in limitinf
    c0, e0 = mrv_leadterm(e, x)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/cache.py", line 94, in wrapper
    retval = cfunc(*args, **kwargs)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/series/gruntz.py", line 522, in mrv_leadterm
    series = calculate_series(f, w, logx=logw)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/series/gruntz.py", line 475, in calculate_series
    for t in e.lseries(x, logx=logx):
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/expr.py", line 2990, in yield_lseries
    for si in s:
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/expr.py", line 3212, in _eval_lseries
    series = self._eval_nseries(x, n=n, logx=logx)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/mul.py", line 1761, in _eval_nseries
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/mul.py", line 1761, in <listcomp>
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/expr.py", line 3289, in nseries
    return self._eval_nseries(x, n=n, logx=logx)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/functions/elementary/exponential.py", line 437, in _eval_nseries
    arg_series = arg._eval_nseries(x, n=n, logx=logx)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/add.py", line 405, in _eval_nseries
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/add.py", line 405, in <listcomp>
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/expr.py", line 3289, in nseries
    return self._eval_nseries(x, n=n, logx=logx)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/functions/special/gamma_functions.py", line 995, in _eval_nseries
    return super()._eval_nseries(x, n, logx)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/function.py", line 685, in _eval_nseries
    return self._eval_aseries(n, args0, x, logx)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/functions/special/gamma_functions.py", line 1011, in _eval_aseries
    return (r + Add(*l))._eval_nseries(x, n, logx) + o
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/add.py", line 405, in _eval_nseries
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/add.py", line 405, in <listcomp>
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/expr.py", line 3289, in nseries
    return self._eval_nseries(x, n=n, logx=logx)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/mul.py", line 1761, in _eval_nseries
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/mul.py", line 1761, in <listcomp>
    terms = [t.nseries(x, n=n, logx=logx) for t in self.args]
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/expr.py", line 3289, in nseries
    return self._eval_nseries(x, n=n, logx=logx)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/power.py", line 1567, in _eval_nseries
    return (1/denominator)._eval_nseries(x, n=n, logx=logx)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/power.py", line 1509, in _eval_nseries
    rest = expand_mul((b - prefactor)/prefactor)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/decorators.py", line 251, in _func
    return func(self, other)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/decorators.py", line 127, in binary_op_wrapper
    return func(self, other)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/expr.py", line 235, in __div__
    return Mul(self, Pow(other, S.NegativeOne))
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/cache.py", line 94, in wrapper
    retval = cfunc(*args, **kwargs)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/operations.py", line 52, in __new__
    c_part, nc_part, order_symbols = cls.flatten(args)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/mul.py", line 197, in flatten
    if not a.is_zero and a.is_Rational:
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/assumptions.py", line 458, in getit
    return _ask(fact, self)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/assumptions.py", line 513, in _ask
    _ask(pk, obj)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/assumptions.py", line 513, in _ask
    _ask(pk, obj)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/assumptions.py", line 501, in _ask
    a = evaluate(obj)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/expr.py", line 861, in _eval_is_positive
    extended_positive = self.is_extended_positive
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/assumptions.py", line 458, in getit
    return _ask(fact, self)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/assumptions.py", line 501, in _ask
    a = evaluate(obj)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/add.py", line 644, in _eval_is_extended_positive
    v = _monotonic_sign(a)
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/exprtools.py", line 150, in _monotonic_sign
    n, d = self.as_numer_denom()
  File "/home/czgdp1807ssd/sympy_project/sympy/sympy/core/add.py", line 505, in as_numer_denom
    denoms, numers = [list(i) for i in zip(*iter(nd.items()))]
KeyboardInterrupt

[czgdp1807]

Closing as the issue is outdated.