Volterra iteration for a singular power-series problem
= Volterra iteration for a singular power-series problem
{title2=$y_n(x)=\sum_{k=0}^n(-1)^kx^{k+1}/[k!(k+1)!]$}
The iteration $y_0=x$, $y_n=x-\int_0^x(x-s)y_{n-1}(s)/s\,ds$ successively appends terms of the displayed series. Induction verifies the iterates, and factorial denominator bounds give convergence with derivatives on compact sets. Its limit is the <power series> solution of $xy''+y=0$, $y(0)=0$, $y'(0)=1$.