Volterra iteration for a singular power-series problem ($y_n(x)={\sum }_{k=0}^n(-1{)}^k{x}^{k+1}/[k!(k+1)!]$)

The iteration $y_0=x$, $y_n=x-{\int }_0^x(x-s)y_{n-1}(s)/sds$ 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 $x{y}^{\prime \prime }+y=0$, $y(0)=0$, $y^{\prime }(0)=1$.