Consider the function defined by
Determine whether the sequence converges or diverges, and if it converges determine its limit.
The sequence diverges.
Proof. First, we use DeMoivre’s theorem to write,
(Note: On page 380 Apostol claims (without proof) that the sequence defined by is divergent. If you want to accept that then this sequence will diverge since a complex-valued sequence diverges if and only if the sequences defined by the functions and converge. Since it is a good exercise (and maybe Apostol wanted us to prove it ourselves) we can prove this is divergent from the definition.)
Suppose, for the sake of contradiction, that the sequence converges to a limit . Then we know for every there exists a positive integer such that for all we have
Since is positive we know and . Hence, taking we have,
Adding these two expressions and using the triangle inequality,
a contradiction. Hence, the sequence diverges