accumulation point unit disc

Dustinsfl

Let [itex]f(z) = \prod\limits_{n = 1}^{\infty}(1 - nz^n) [/itex]

Prove that each point on the unit circle is an accumulation point of zeros of [itex]f [/itex]

So we have that [itex]z = \sqrt[n]{1/n} [/itex]. Now where do I go from here?

Probably should note that this is a Weierstrass Product.

sunjin09

The set of all zeros in of f(z) is [itex]\{\sqrt[n]{1/n} e^{i2\pi\frac{k}{n}}|n,k\in Z_+\}[/itex], now for any [itex]z=e^{i\phi}[/itex] on unit circle, there exit n, k such that [itex]\sqrt[n]{1/n} e^{i2\pi\frac{k}{n}}[/itex] is close enough to [itex]z=e^{i\phi}[/itex] in both amplitude and phase ...

Dustinsfl

By phase, you mean argument?

sunjin09

yes, I'm an electrical engineer :)

Dustinsfl

Ok thanks. That problem was relatively easy.