Malus' law in the limit of infinitely many polarizers

[greypilgrim]

Hi.

How can I prove
$$\lim_{n\to\infty} \cos(\alpha/n)^{2n}=1$$
for all ##\alpha\in\mathbb{R}##? The physical background is Malus' law for perfect linear polarizers, I'd like to show that one can losslessly rotate a linearly polarized wave by any angle by stacking an infinite number of infinitely rotated polarizers.

 

[mathman]

Rough proof: [itex](cos(\alpha /n))^{2n}\approx (1-\frac{\alpha^2}{2n^2})^{2n}\approx e^{-\frac{\alpha^2}{n}} \to 1[/itex]

 

[zaidalyafey]

You can take the log and use L'H rule.