Consider the nonhomogeneous linear equation $y' = 2y/(1-x) + f(x)$. It is singular at $x=1$, of course, but it is regular at $x=0$ if (the known function) $f(x)$ is analytic there. Assume $y(x) = \sum_{m=0}^\infty a_mx^m$ and $f(x) = \sum_{m=0}^\infty b_mx^m$.
(d) Substitute these series directly into the differential equation and, recalling the geometric series $1/(1-x) = \sum_{m=0}^\infty x^m$, derive the recursion relation $a_n =\frac2n\sum_{r=0}^{n-1}a_r + b_{n-1}/n$.
I'm having some trouble to derive the recursion relation shown in the screenshot.