Suppose that $S(x) = \sum^{∞}_{k=0}{a_k x^k}$ is a power series such that $|\frac{a_n +1}{a_n}|→ λ > 0 $ as $n → ∞$. What is the radius of convergence of $S$? (The value $∞$ for $λ$ is permitted. You don’t need to justify your answer.)
Answer:
If $|\frac{a_n +1}{a_n}|→ λ > 0 $ then the radius of convergence $R$ equals $R= \lambda^{-1}$, with the convention $∞^{-1}=0$.
Is this correct? It it the answer given in the past paper solutions, but I don't really understand the last part of "with the convention $∞^{-1}=0$".
It does sound like the actual definition of radius of convergence, and the reciprocal of infinity being zero, is to say that, that series would only converge at x= 0.