Are Cycles good expanders???

Question

The definition of a $k$-regular Ramanujan graph is that $\mu_1$, the largest non-trivial (meaning not equal to $k$) is less than or equal to $2\sqrt{k-1}$.
All the eigenvalues of $C_n$ are between $-2$ and $2$ and thus the non-trivial ones are less than $2=2\sqrt{2-1}$, so the family $C_n$ of cycles makes for a family of Ramanujan graphs. Allegedly Ramanujan graphs make for good expanders, that is their Cheeger Constants are high, or in other words they are sparse and highly connected. However cycles are not highly connected, which seems to be a counter example... where am I going wrong?