Is Every Point on the Unit Circle an Accumulation Point of Zeros for f(z)?

[Dustinsfl]

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

Prove that each point on the unit circle is an accumulation point of zeros of f

So we have that z = \sqrt[n]{1/n}. 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 \{\sqrt[n]{1/n} e^{i2\pi\frac{k}{n}}|n,k\in Z_+\}, now for any z=e^{i\phi} on unit circle, there exit n, k such that \sqrt[n]{1/n} e^{i2\pi\frac{k}{n}} is close enough to z=e^{i\phi} in both amplitude and phase ...

 

[Dustinsfl]

By phase, you mean argument?

 

[sunjin09]

yes, I'm an electrical engineer :)

 

[Dustinsfl]

yes, I'm an electrical engineer :)

Ok thanks. That problem was relatively easy.