Let $D$ be the derivative operator, and set $$T:=\sum_{n=0}^\infty D^n. $$ It is known that under certain conditions $$T=(Id-D)^{-1} .$$ Set $v=Tu$, applying $I-D$ to both sides, in this case, gives the ODE $$v-v'=u $$ which can be solved for $v$ as follows $$v=\mathrm{e}^x \left( C-\int \mathrm{e}^{-x} u(x) \mathrm{d} x \right),$$ where $C$ is a constant of integration, originating from the indefinite integral. At the same time we also have $$v=Tu=\sum_{n=0}^\infty u^{(n)}(x), $$ so we seem to get $$ \mathrm{e}^x \left( C-\int \mathrm{e}^{-x} u(x) \mathrm{d} x \right)=\sum_{n=0}^\infty u^{(n)}(x).$$ The left hand side is an infinite 1-parametric family of functions, while the right hand side is a concrete function, provided the series converges (e.g. the case where $u(x)$ is a polynomial).
My questions are: how does one settle this issue? Is the integral on the left hand side satisfying some initial/boundary condition? What is the value of $C$?
Thank you!