A recursive sum

Question

Let's say we start with the number one, then we add $(1/1)$. Now we have $2$. Then we do $2 + (1/2)$. Now we have $2.5$. Then we do $2.5$ plus $(1/2.5)$. If we continue this forever, I am fairly sure this diverges. However, I am wondering whether there is an implicit formula for the $n$th term of this sequence. For example: $A(1)=1$, $A(2)=2$, $A(3)=2.5$, and so on.