Is it true that $\sum\limits_{n=1}^N e^{in} = \frac{e^i(1-e^{iN})}{1-e^i}$ even if $| e^i | > 1$?

Question

Is it true that $\sum\limits_{n=1}^N e^{in} = \frac{e^i(1-e^{iN})}{1-e^i}$ even if $| e^i | > 1$?

I know this question is quite trivial and I will understand if it gets removed.

I am trying to show that $\sum\limits_{n=1}^\infty \frac{cos(n)}{\sqrt{n}}$ is convergent by using the Dirichlet's test. And taking, for the one series, $cos(n)$, now

$|\sum\limits_{n=1}^N cos(n) |< |\sum\limits_{n=1}^N (cos(n) + i sin(n))| = |\sum\limits_{n=1}^N e^{in}|$

And I am trying to find a value for $|\sum\limits_{n=1}^N e^{in}|$

Answer 1

Yes, the statement $$ \sum\limits_{n=1}^N e^{in} = \frac{e^i(1-e^{iN})}{1-e^i} $$ for any choice of $e \in \Bbb C$. The only issue you have is whether the limit exists when $N \to \infty$.

Note that the proof (of this and the general geometric series formula) is not particularly difficult. We have $$ \sum\limits_{n=1}^N e^{in} = e^{i} + e^{2i} + \cdots + e^{iN}\\ e^i\sum\limits_{n=1}^N e^{in} = e^{2i} + \cdots + e^{iN} + e^{i(N+1)} $$ Subtract the two, factor, and solve.

Answer 2

$|e^{it}|$ is always equal to $1$, for any $t$. To see this note that $e^{it}\overline{e^{it}}=e^{it}e^{-it}=1$. Now, if a complex number $z$ satisfies $z\overline{z}=1$, then by definition, $|z|^2=1$ and hence $|z|=1$.