Splitting $2n + 1$ cards, what is the expected ratio of a size of the smaller deck to the bigger deck?

Assuming equal probability of the deck where the split is.

I can find the expected size of the smaller deck of card is $\frac{1+n}{2}$, but I am not sure how the find the expected ratio in a closed form of $n$.

Fundamentally, it would be something like $\frac{1}{n}\sum_{i=1}^n \frac{i}{2n+1-i}$. Not sure if there is a closed form. Maybe there is a different approach.