How do we compute the $k$-th rational homology group of the connected sum $\#^{\infty} S^{\infty}$, and hence the sequence of rational Betti numbers ${b_k\left(\#^{\infty} S^{\infty};\mathbb{Q}\right)}=\text{rk} H_k\left(\#^{\infty} S^{\infty};\mathbb{Q}\right)$? Since finding the homology of a sphere is trivial, I am essentially asking how to find the homology of a union of topological spaces?
Context: The space under consideration is embedded in $\mathbb{R}^{\infty}$. In particular, if we consider an infinite-dimensional Hilbert manifold $M$, then we patch it via an atlas $\bigcup_{\alpha\in A}\mathcal{U}_{\alpha}$ for $A$ countable, but infinite and where $\mathcal{U}_{\alpha}$ is $S^{\infty}$,$\forall\alpha\in A$.
Thanks in advance!