Need help understanding the concept
Let $z = e^{i\theta} \in S^1$ , $S^1 = \{z:|z|=1\}$
Suppose z is a root of 1, i.e, $z=e^{2\pi i p/q}$ , $p, q \in \mathbb Z, q0$
For $n\geq q$ q | n! and $z^{n!}=1$
Next is what I don't get. (continue on comment)

Question

Need help understanding the concept
Let $z = e^{i\theta} \in S^1$ , $S^1 = \{z:|z|=1\}$
Suppose  z is a root of 1, i.e, $z=e^{2\pi i p/q}$ , $p, q \in \mathbb Z, q>0$
For $n\geq q$ q | n! and $z^{n!}=1$
Next is what I don't get. (continue on comment)

loser66 · Answer

So, 
$\sum_{n =1}^\infty \dfrac{(-1)^n}{n}z^{n!}=\sum_{n=1}^{q-1} \dfrac{(-1)^n}{n}z^n +\sum_{n =q}^{\infty} \dfrac{(-1)^n}{n}$

loser66 · Answer

The first sum is finite and so does not affect convergence
The second one is an alternating harmonic series which converges. Then the whole thing converges. 
I don't get it. Please, help how can the sum be break into 2 parts like that?

loser66 · Answer

@ganeshie8

loser66 · Answer

nvm, I got it.