Loser66
  • Loser66
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)
Mathematics
chestercat
  • chestercat
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Loser66
  • Loser66
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
  • Loser66
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
  • Loser66

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Loser66
  • Loser66
nvm, I got it.

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