it's known that if $ g(x), f(x)$ are two functions ,and $f(x)$ is sufficiently differentiable , then by repeated integration by parts one gets :
\int f(x)g(x)dx=f(x)\int g(x)dx -f^{'}(x)\int\int g(x)dx^{2}+f^{''}(x)\int \int \int g(x)dx^{3} - .... +(-1)^{n+1}f^{(n)}(x)\underbrace{\int.....\int}g(x)dx^{n+1}+(-1)^{n}\int\left[ \underbrace{\int.....\int}g(x)dx^{n+1}\right ]f^{n+1}(x)dx
now, if $f(x) $ is a smooth function,and none of the terms in the expansion/summation is equal to $\pm\int f(x)g(x)dx$ , one would expect the formula above to be repeatable infinitely many times . therefore :
$\lim_{n \to \infty }\int\left[ \underbrace{\int.....\int}g(x)dx^{n+1}\right ] f^{n+1}(x)dx=0$
is a necessary but not sufficient condition for the summation to converge . my question is , what are the conditions needed to extend the scope of the formula - to perform the IBP infinitely many times - !?!? also, are there any theorems on the multiple integrals - the ones containing $g(x)$ - besides cauchy formula for repeated integration