This shows that product cannot tend to

(i.e.,

).

On the other hand, we have

showing that the product can not also diverge (in the sense of complex modulus).

Then, the complex sequence

lies in the annulus

.

Due to Bolzano-Weierstrass theorem (considered separetely for the real and imaginary parts),

there exists a subsequence

of

which converges to some

.

Recall that

and

,

which shows that

for

.

Therefore, the product accumulates to at least (also at most) four different points

since

.

But I just wanted to know if it is possible to find (by hand) what

is.