I am asked to tackle the following problem:
"Prove that if and Re 0 for all k = 1, 2, 3... then "
My first issue is one of interpretation. What am I looking to show? That if z belongs to [0, ), then z must be purely real? I don't know too much about the where complex numbers fit into the typical set notation. How would I attempt this problem and what should a starting point be?
What I have attempted so far is to multiply out a few of the lower powers of z and collect real and imaginary parts, but that doesn't seem to get me anywhere of use. The real part of z must be nonnegative from the definition, taking k=1. Then, from k=2 we can see that the real part of z must be greater than the imaginary part, then with k=3, the real part is greater than 3 times the imaginary part. By trial and error, I can keep increasing k and see how this is going but its not rigorous. Is this even the right approach, and if so should I be using a proof by induction?
Thanks for your reply,
If I am understanding correctly, I think this implies has to be zero.
I think I am getting the picture now. I can see why z has to be purely real, I can convince myself; but I don't know how I would prove it from this.