# Error in perturbation series question

Suppose we know the perturbation series

$$E = E(\epsilon) = E_0 + \epsilon E_1 + \epsilon^2 E_2 + \ldots$$

converges, where $E_0$ is a discrete eigenvalue of $H_0$ and we are considering a Hamiltonian $H = H_0 + \epsilon H_1$. Does this mean that we know

$$E - E_0 = O(\epsilon)$$

as $\epsilon \to 0$ in the precise sense that we know there exists a $\delta > 0$ and a $C > 0$ such that if $|\epsilon| < \delta$, then $|E - E_0| \leq C|\epsilon|$?