This question was asked in my complex analysis assignment and I was unable to solve it. So I am posting here as assignement might not will be discussed.
Find the region of Convergence for $\prod_{n=1}^{\infty}\cos(z/n) $.
Attempt: I wrote the given product as $\prod_{n=1}^{\infty} ( 1-(1-\cos(z/n)) $. The term $\lim_{n\to \infty}|( 1-(1-\cos(z/n)) | \to 0$ ( If question was about $\sin(z/n)$ then the corresponding sum would go to 1 and from that it could be deduced that Corresponding product of $\sin(z)$ would diverge everywhere. )
But , this argument certainly can't be used for cos and I am unable to think of any other result which I can use. So, please help.
Thanks!