= Picard iteration for a simple-pole linear differential equation
{c}
{title2=$y'=ay/(1-x),\ y(0)=c\implies y_n=c\sum_{k=0}^n[-a\log(1-x)]^k/k!$}
For real constants $a,c$, apply <Picard iteration> to the <initial value problem> $y'=ay/(1-x)$, $y(0)=c$, starting with $y_0(x)=c$. The recurrence is
$$
y_{n+1}(x)=c+\int_0^x\frac{a\,y_n(t)}{1-t}\,dt.
$$
Since $\int_0^x\log^k(1-t)/(1-t)\,dt=-\log^{k+1}(1-x)/(k+1)$, <mathematical induction> gives the displayed partial sums. The <exponential series> converges at every finite argument, so $y_n(x)\to c(1-x)^{-a}$ for every real $x<1$, uniformly on each <compact interval> in that range. This is not a <Taylor series> in $x$, so its convergence is not restricted to $|x|<1$. The differential equation's coefficient is singular at $x=1$ even when a special solution has a removable singularity there.
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