A theoretical question regarding Frobenius method

Question

The following is a theoretical question regarding Frobenius method. Let $b(x),c(x)$ be real functions analytic at 0. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. Suppose $r$ is a root of $r(r-1)+b_0r+c_0=0$ such that $r+k$ is not a root for any positive integer $k$. Then for $a_0=1$, the recurrence relation $$a_k((r+k)(r+k-1)+b_0(r+k)+c_0)=-\sum_{i=1}^k(b_i(r+k-i)+c_i)a_{k-i}, k\ge 1$$ defines a sequence $a_k$. Let $S$ be the radius of convergence of $$\sum_k a_kx^k.$$ In G. Birkhoff and G.C. Rota's Ordinary Differential Equation, 4th ed., Chapter 9, section 8, Theorem 7, it is shown that $S>0$. My question is whether $S$ can be strictly less than $R$?