If a finite sum has a definite integral representation, for which it can be proved the underlying *indefinite* integral is not an elementary function, then does this imply the original finite sum can not be expressed as an elementary function (on applying the bounds of the original representation). Or does this mean that the method of integration is a dead end in this case?

I suppose another away of asking this question is: Do there exist finite sums which can be expressed as an elementary function (without summation signs), but for which there also exist integral representations whose associated indefinite integral is non-elementary?