Worked mathematical example B07 — subtraction and a factorial series

Substituting ${\mu }_R=\mu$ in [F1] gives ${\mu }^2\log (1)\mathrm{/}(16{\pi }^2)=0$, so the subtracted coefficient tends to zero. At the same time, [F1] gives $I_{\mathrm{\Lambda }}\mathrm{/}{\mathrm{\Lambda }}^2\to 1\mathrm{/}(16{\pi }^2)>0$. Since ${\mathrm{\Lambda }}^2\to \mathrm{\infty }$, this implies $I_{\mathrm{\Lambda }}\to +\mathrm{\infty }$ and records its quadratic leading coefficient. The two statements concern different quantities: one is scheme-subtracted and the other is the positive bare cutoff integral.

For every fixed integer $n\ge 0$, the integral moment ${\int }_0^{\mathrm{\infty }}{x}^n\exp (-x)dx=n!$ is finite by [F1]. But for each $\lambda >0$, consecutive absolute terms of the formal series have ratio $\lambda (n+1)$, which eventually exceeds $2$; therefore its terms do not tend to zero and the series diverges. The exact finite remainder in [F1] does not turn that divergent infinite series into a convergent sum for $\lambda >0$. Nothing here measures fluctuations or proves a continuum limit.