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You defined

sage: a = sum(sqrt(x), x, 0, oo); a
sum(sqrt(x), x, 0, +Infinity)

Let us explore this object.

sage: a.parent()
Symbolic Ring

Typing a. and pressing the tab key, we find among other methods the following two.

sage: a.operator()
sum
sage: a.operands()
[sqrt(x), x, 0, +Infinity]

So we can define a function that will replace +Infinity by n:

sage: def partial(a,n):
....:     return a.operator()(*(a.operands()[:-1]+[n]))
....: 
sage: partial(a,10)
sum(sqrt(x), x, 0, 10)

and use it to get the first few partial sums:

sage: for n in xrange(5r):
....:     print partial(a,n)
....:     
sum(sqrt(x), x, 0, 0)
sum(sqrt(x), x, 0, 1)
sum(sqrt(x), x, 0, 2)
sum(sqrt(x), x, 0, 3)
sum(sqrt(x), x, 0, 4)