Radius of convergence of Eisenstein series

Question

The Eisenstein series $E_k$ on the upper half plane $\mathbb H$ are holomorphic, $1$-periodic and therefore allow a Fourier series $$E_k(\tau)=1-\frac{2k}{B_k}\sum_{n=1}^{\infty}\sigma_{k-1}(n)q^n, \quad q=e^{2\pi i\tau},$$ with $k\geq2$ even, $B_k$ the Bernouilli numbers and $\sigma$ the divisor function. This is also the Taylor series at $\tau=i\infty$ or $q=0$, respectively. Let us consider $E_4$ as an example. In order to check the radius of convergence, we can use the ratio test. If the series converges on the whole $\mathbb H$, the radius of convergence has to be at least 1. However, one can check numerically that $\frac{\sigma_3(n+1)}{\sigma_3(n)}$ does not approach 1 for $n\to\infty$. Does that mean the $q$-series of $E_4$ does not converge on the whole $\mathbb H$ or is there a flaw in my argument? Thanks a lot for a quick answer!