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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.