Construct a sequence of complex numbers $(z_n)_{n \geq 1}$ such that $\sin z_n$ is real for all $n$ and $$\lim_{n \to \infty} \sin{z_n}=\infty$$. Or give a proof that there exist no such sequence.
2026-04-28 19:21:32.1777404092
Sequence of complex numbers for which sine blows up
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Just solve the equation $\sin (z_n)=n$. If $w=e^{iz_n}$ this equation becomes $w-\frac 1 w =2ni$ or $w^{2}-2niw-1=0$. You have to solve this quadratic and then take logarithm. (There are infinitely many choices for $z_n$!).