Comment #14 on issue 1092 by [email protected]: limit(sum(1/k, (k, 1, n))-log(n), n, oo) should do it
http://code.google.com/p/sympy/issues/detail?id=1092

Currently, it returns the correct limit

from sympy import *
k = Symbol('k')
n = Symbol('n')
limit(summation(1/k, (k, 1, n))-log(n), n, oo)
EulerGamma

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