Make limit work with binomial

[asmeurer]

limit can work with binomial by converting to factorial first.

In [214]: limit(16**k/(k*binomial(2*k,k)**2), k, oo)
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
/Users/aaronmeurer/Documents/Python/sympy/sympy/sympy/series/limits.py in doit(self, **hints)
    171         try:
--> 172             r = gruntz(e, z, z0, dir)
    173             if r is S.NaN:

/Users/aaronmeurer/Documents/Python/sympy/sympy/sympy/series/gruntz.py in gruntz(e, z, z0, dir)
    640     if z0 == oo:
--> 641         r = limitinf(e, z)
    642     elif z0 == -oo:

/Users/aaronmeurer/Documents/Python/sympy/sympy/sympy/series/gruntz.py in limitinf(e, x)
    423         x = p
--> 424     c0, e0 = mrv_leadterm(e, x)
    425     sig = sign(e0, x)

/Users/aaronmeurer/Documents/Python/sympy/sympy/sympy/series/gruntz.py in mrv_leadterm(e, x)
    479     if Omega == SubsSet():
--> 480         Omega, exps = mrv(e, x)
    481     if not Omega:

/Users/aaronmeurer/Documents/Python/sympy/sympy/sympy/series/gruntz.py in mrv(e, x)
    258         s1, e1 = mrv(a, x)
--> 259         s2, e2 = mrv(b, x)
    260         return mrv_max1(s1, s2, e.func(i, e1, e2), x)

/Users/aaronmeurer/Documents/Python/sympy/sympy/sympy/series/gruntz.py in mrv(e, x)
    257         a, b = d.as_two_terms()
--> 258         s1, e1 = mrv(a, x)
    259         s2, e2 = mrv(b, x)

/Users/aaronmeurer/Documents/Python/sympy/sympy/sympy/series/gruntz.py in mrv(e, x)
    265         else:
--> 266             s, expr = mrv(b, x)
    267             return s, expr**e

/Users/aaronmeurer/Documents/Python/sympy/sympy/sympy/series/gruntz.py in mrv(e, x)
    294             # e.g. something like BesselJ(x, x)
--> 295             raise NotImplementedError("MRV set computation for functions in"
    296                                       " several variables not implemented.")

NotImplementedError: MRV set computation for functions in several variables not implemented.

During handling of the above exception, another exception occurred:

NotImplementedError                       Traceback (most recent call last)
<ipython-input-214-830bdb8908f4> in <module>()
----> 1 limit(16**k/(k*binomial(2*k,k)**2), k, oo)

/Users/aaronmeurer/Documents/Python/sympy/sympy/sympy/series/limits.py in limit(e, z, z0, dir)
     43     """
     44
---> 45     return Limit(e, z, z0, dir).doit(deep=False)
     46
     47

/Users/aaronmeurer/Documents/Python/sympy/sympy/sympy/series/limits.py in doit(self, **hints)
    183                 r = limit_seq(e, z, trials)
    184                 if r is None:
--> 185                     raise NotImplementedError()
    186             else:
    187                 raise NotImplementedError()

NotImplementedError:

In [215]: limit(16**k/(k*binomial(2*k,k)**2).rewrite(factorial), k, oo)
Out[215]: π

[wrat]

Means we have to convert binomial to factorial in limit.py ??but how it can recognize that given expression is binomial??

[asmeurer]

You can use expr.has(binomial).

[wrat]

ohkkk , I am looking into this .

[wrat]

gruntz fails on factorials but works with the gamma function .so
e= e.rewrite(factorial) or e = e.rewrite(factorial,gamma) ??
I have tried from e=e.rewrite(factorial) then it is showing 'pi' as output (which is right)."
But why it shows not implemented error using e=e.rewrite(factorial,gamma) .

[wrat]

@asmeurer
def test_limit_seq(): assert (limit(binomial(2*x, x) / Sum(binomial(2*y, y), (y, 1, x)), x, oo) == S(3) / 4)
can we discuss about this test case??

[leosartaj]

@wrat This limit is found using a different algorithm, implemented here. It is currently called as last resort, in-case no other algorithm can find the limit. Reason being, it is still new, and capable of solving only a small subset.

[wrat]

See PR #10825

[asmeurer]

I think we can also fix this by implementing binomial._rewrite_as_tractable.

[wrat]

Yeah of course we can fix it like that but while implementing it , may be we will get some issue according to me actually .so should I implement like binomial._rewrite_as_tractable to get merge.

[wrat]

@asmeurer ping

[asmeurer]

I think the as_tractable way is the best way to fix this.