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,'real') doesn't work, as I get the same error...how can I fix all this?
Thanks in advance!
