= Characteristic ratio of a freely rotating chain
{title2=$C_N=\frac{1+\alpha}{1-\alpha}-\frac{2\alpha(1-\alpha^N)}{N(1-\alpha)^2},\quad\alpha=\cos\theta$}
For $N$ bonds of length $a$, fixed forward angle $\theta$ and uniform independent torsions, conditional averaging gives $\langle\mathbf a_i\cdot\mathbf a_j\rangle=a^2\alpha^{|j-i|}$. Consequently $\langle R^2\rangle=Na^2+2a^2\sum_{k=1}^{N-1}(N-k)\alpha^k$. Dividing by the independent-step result $Na^2$ and summing the <geometric series> gives the displayed $C_N$, for $\alpha\ne1$. For fixed $|\alpha|<1$, $C_N\to C_\infty=(1+\alpha)/(1-\alpha)$.
The angle is measured between forward vectors; a chemical internal bond angle is supplementary. At $\alpha=1$, the finite chain is a rigid rod with $C_N=N$, so taking the infinite-chain limit before the rigid limit would be invalid. The <Kuhn length of a freely rotating chain> is asymptotically $aC_\infty$.
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