Homework Help: Questions on Complex Analysis

1. Sep 27, 2012

raopeng

1. The problem statement, all variables and given/known data
To prove that the sequence $a_{n}= \prod_{k}^\infty (1 + \frac{i}{k})$ when n is infinite constitutes points on a circle.

2. Relevant equations
Ehh no idea what equations shall be used.

3. The attempt at a solution
A friend asked me this, but I am usually engaged more with the physical aspects of Complex Analysis... so I have no idea how I should approach this question.

2. Sep 27, 2012

Studiot

Well points on a circle have a particular (finite) radius.

So you need to prove your infinite product has a finite limit,
Since we are in the complex plane this will be a radius, as every point with this modulus will be included.

3. Sep 28, 2012

raopeng

Thank you. Yes I noticed that too, as we can extract an infinite series from it...

4. Sep 28, 2012

Studiot

So what happens if you write a few of the terms of the series out and multiply them in pairs?

Further hint put 1/k = α.
So the terms take the form (1+αi)

5. Sep 28, 2012

raopeng

Thank you for the help. Also $arg a_{n} = \sum^{\inf}_{k}\frac{1}{k}$ covers the entire circle.