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Problem evaluating limit: assume doesn't work!

asked 2014-12-14 21:22:50 -0500

nespinoza gravatar image

Hi everyone,

I'm trying to evaluate the following limit in sage:

$$\lim_{N\to \infty}\frac{6N^k(N-1)^{1-k}}{(2N-1)(k+1)},$$

where $k\neq \pm1$. From Wolfram-Alpha I know this is equal to $3/(1+k)$, but I was trying to get to that answer with sage with no progress. What I did to do it was the following:

sage: N = var('N')
sage: k = var('k')
sage: assume(k!=1)
sage: assume(k!=-1)
sage: limit((6*N^k*(N-1)^(1-k))/((2*N-1)*(k+1)),N=oo)

But I get the following error:

ValueError                                Traceback (most recent call last)
<ipython-input-7-bb63a0377d1b> in <module>()
----> 1 limit((Integer(6)*N**k*(N-Integer(1))**(Integer(1)-k))/((Integer(2)*N-Integer(1))*(k+Integer(1))),N=oo)

/usr/lib/sagemath/local/lib/python2.7/site-packages/sage/calculus/calculus.pyc in limit(ex, dir, taylor, algorithm, **argv)
   1198     if algorithm == 'maxima':
   1199         if dir is None:
-> 1200             l = maxima.sr_limit(ex, v, a)
   1201         elif dir in ['plus', '+', 'right', 'above']:
   1202             if dir == 'above':

/usr/lib/sagemath/local/lib/python2.7/site-packages/sage/interfaces/maxima_lib.py in sr_limit(self, expr, v, a, dir)
    855                 j = s.find('Is ')
    856                 s = s[j:]
--> 857                 raise ValueError, "Computation failed since Maxima requested additional constraints; using the 'assume' command before limit evaluation *may* help (see `assume?` for more details)\n" + s
    858             else:
    859                 raise error

ValueError: Computation failed since Maxima requested additional constraints; using the 'assume' command before limit evaluation *may* help (see `assume?` for more details)
Is  k-1.0  positive or negative?

In general it doesn't matter if $k$ is positive or negative, they converge to the same value. However, for the sake of completeness, let's first try to restrict $k$ to values between 0 and 1. So, I opened other sage session and did:

sage: N = var('N')
sage: k = var('k')
sage: assume(k>0,k<1)
sage: limit((6*N^k*(N-1)^(1-k))/((2*N-1)*(k+1)),N=oo)

But I got:

ValueError                                Traceback (most recent call last)
<ipython-input-4-bb63a0377d1b> in <module>()
----> 1 limit((Integer(6)*N**k*(N-Integer(1))**(Integer(1)-k))/((Integer(2)*N-Integer(1))*(k+Integer(1))),N=oo)

/usr/lib/sagemath/local/lib/python2.7/site-packages/sage/calculus/calculus.pyc in limit(ex, dir, taylor, algorithm, **argv)
   1198     if algorithm == 'maxima':
   1199         if dir is None:
-> 1200             l = maxima.sr_limit(ex, v, a)
   1201         elif dir in ['plus', '+', 'right', 'above']:
   1202             if dir == 'above':

/usr/lib/sagemath/local/lib/python2.7/site-packages/sage/interfaces/maxima_lib.py in sr_limit(self, expr, v, a, dir)
    855                 j = s.find('Is ')
    856                 s = s[j:]
--> 857                 raise ValueError, "Computation failed since Maxima requested additional constraints; using the 'assume' command before limit evaluation *may* help (see `assume?` for more details)\n" + s
    858             else:
    859                 raise error

ValueError: Computation failed since Maxima requested additional constraints; using the 'assume' command before limit evaluation *may* help (see `assume?` for more details)
Is k an integer?

Trying to do sage: assume(k ... (more)

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answered 2014-12-15 03:58:40 -0500

tmonteil gravatar image

Assuming that k is real is much weaker than assuming that k is an integer. Also it seems that you need to simplify the expression first to let the computation work:

sage: N = var('N')
sage: k = var('k')
sage: assume(k-1>0)
sage: assume(k, 'integer')
sage: E = (6*N^k*(N-1)^(1-k))/((2*N-1)*(k+1))
sage: limit(E.full_simplify(),N=oo)
3/(k + 1)
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Hi @tmonteil; thanks for your anwer. I understand that assuming that k is real is much weaker than assuming that k is an integer, but, as I implicitly showed, I'm using this for non-integers inclusive.

Also, I'm using Sage Version 6.1.1 (Release Date: 2014-02-04) and this doesn't work; it gives me the same error I showed, and asks me: "Is k-1 positive, negative, or zero?" (which is strange since I already defined it to be positive).

nespinoza gravatar imagenespinoza ( 2014-12-15 06:44:04 -0500 )edit

Yeah, Maxima's assumptions are notoriously weak. We can of course report something upstream if you can confirm with the most recent Sage.

kcrisman gravatar imagekcrisman ( 2014-12-15 09:46:29 -0500 )edit

I'm trying to update my Sage version to try this but it won't let me; do you know if there's something going on with the trac server? I try: sage -upgrade, and it gives me:

Upgrading to the latest development version fatal: unable to connect to trac.sagemath.org: trac.sagemath.org[0: 128.208.178.249]: errno=Connection refused

nespinoza gravatar imagenespinoza ( 2014-12-15 10:28:16 -0500 )edit

Ok, solution works with new version, but still think that imposing $k$ to be an integer is too much...

nespinoza gravatar imagenespinoza ( 2014-12-15 12:49:04 -0500 )edit

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Asked: 2014-12-14 21:22:50 -0500

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Last updated: Dec 15 '14