 Solution

For $0<K<\infty$, $|q|=\tanh K<1$. Taking the thermodynamic limit $N\to \infty$ at fixed separation $n$ removes $q^N$ and $q^{N-n}$ from the finite antiferromagnetic Ising-ring correlation. ThereforeThe correlation length $\xi$ is measured in lattice spacings. If the lattice spacing is $a$, its physical value is $a\xi$. The factor $(-1{)}^n$ expresses antiferromagnetism, while $e^{-n/\xi }$ measures the decay of the staggered two-point correlation function.

At every positive temperature, $\xi$ is finite. At low temperature, $\tanh K=1-2e^{-2K}+O(e^{-4K})$, giving $\boxed{\xi \sim \frac{1}{2}{e}^{2\beta J}}$. At infinite temperature, $K=0$, distinct Ising spins are uncorrelated; the limiting correlation length is zero.