Why can we crank up the variable $\lambda$ in perturbation theory?

Question

In the formulation of perturbation theory, we use the power series (more specifically geometric series) for small expansion, say $\lambda$, to expand the exact case, say $H^0$, so $\begin{aligned} & E_n=E_n^{(0)}+\lambda E_n^{(1)}+\lambda^2 E_n^{(2)}+\cdots \\ & |n\rangle=\left|n^{(0)}\right\rangle+\lambda\left|n^{(1)}\right\rangle+\lambda^2\left|n^{(2)}\right\rangle+\cdots \end{aligned}$ . Now; later this expansion variable is 'cranked up' to 1. How does the expansion still reflect the exact case of $H^0$ ? Is the power series not defined by small $\lambda$ ? How do we know if this infinite series converges? Does the infinite series 'imply' that H possesses the necessary properties for convergence?