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Sage doesn't show limit of ratio of sums

asked 2014-11-02 16:23:58 -0600

nespinoza gravatar image

Hi everyone,

I'm new to Sage and I'm trying to obtain the following limit: $$\lim_{N\to \infty}\frac{\sum_{i=1}^{N} x_i^{5/2}}{\sum_{i=1}^N x_i^2}$$ with $x_i=(N-i)/(N-1)$. What I tried was:

i = var('i')
assume(i>0,'integer')
N = var('N')
assume(N>0,'integer')
x_i = (N-i)/(N-1)
numerator = sum(x_i^(5./2.),i,1,N)
denominator = sum(x_i^(2.),i,1,N)
limit(numerator/denominator,N=oo)

However, this just gives me:

6*limit((N - 1)*sum(((N - i)/(N - 1))^2.5, i, 1, N)/(2*N^2 - N), N, +Infinity)

From what I can see, Sage has no problems with the denominator, the general formula of which was obtained easily. However, apparently it has problems obtaining the general formula for the numerator due to the fact that the exponent in the sum is not an integer. Is there a way to cope with this? How can I obtain the limit?

Thanks in advance for all the help!

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answered 2014-11-03 11:46:57 -0600

Sage uses Maxima for formal sums and so depends on that package. Maxima can compute

sage: sum(1/i^3,i,1,N)
gen_harmonic_number(3, N)

but unfortunately Sage cannot use this result symbolically. A ticket for this is waiting for review: http://trac.sagemath.org/ticket/16671

However, to compute your sum Maxima would need to know about generalized harmonic numbers with rational index. So this functionality must first be implemented in Maxima. Even then, this won't give you the limit which, I think, could be computable with some holonomic package. Something like this is missing in Sage, at the moment, as well.

Finally, I could well be wrong in the above, and there may be a special shortcut for implementation.

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I see...well, I'll have to wait then. Simulations are good enough anyways to actually see the limiting value!

nespinoza gravatar imagenespinoza ( 2014-11-03 13:29:44 -0600 )edit

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Asked: 2014-11-02 16:23:58 -0600

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Last updated: Nov 03 '14