Proving the Circle Property of Infinite Sequence in Complex Analysis

[raopeng]

Homework Statement

To prove that the sequence $a_{n}= \prod_{k}^\infty (1 + \frac{i}{k})$ when n is infinite constitutes points on a circle.

Homework Equations

Ehh no idea what equations shall be used.

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.

 

[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.

 

[raopeng]

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

 

[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)

 

[raopeng]

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