Definite integrals

The notion of limiting sum cannot be expressed as a signature. There is no notion of approximation with signatures, arbitrarily close rational numbers can have arbitrarily different signatures. Consequently, under our present knowledge of signatures, it does not appear possible to compute signatures of definite integrals, definite sums, limits and similar operations. Additionally, integrals or summations with infinite limits are completely outside the reach of signatures for an even stronger reason.