Let $\{x_n\}_{n\in\mathbb{N}}$ be a family of elements of the Hilbert spaces $H_n$.
Suppose that $\underset{n\rightarrow+\infty}{\lim}\parallel x_n\parallel_n^2=0$. It seems reasonable to me that we can therefore say that there exists $N\in\mathbb{N}$ such that $$ \underset{n=N+1}{\overset{\infty}{\sum}}\parallel x_n \parallel_n^2<\epsilon^2 $$ for any $\epsilon>0$.
However, I am not sure of this, and I am failing to see how one can prove it, if it is correct.
Consider $H_n = \mathbb R^n$, and give each $H_n$ the Euclidean norm. Consider $x_n = (1/n,1/n,\ldots,1/n)$. Then $\|x_n\|_{H_n}^2 = \frac1{n}$ for each $n$. So $\sum \|x_n\|^2$ is the harmonic series - it diverges.