Investigating Complex Series: Convergence or Divergence?

[ehrenfest]

Homework Statement

Investigate the behavior (convergence or divergence) of \sum_n 1/(1+z^n) where z is complex.

Homework Equations

The Attempt at a Solution

If the modulus of z is less than 1, it is not hard to show that the limit of the sequence is not 0 (it is actually not finite) and thus the series cannot converge. But if the modulus of z is greater than or equal to 1, I don't what to apply. The root test? The ratio test? The comparison test?

 

[e(ho0n3]

Would you care to show how the series diverges if |z| < 1?

I determined that if |z| > 1, then series converges. I did this by comparing |zⁿ| and |1 + zⁿ| and using the comparison test.

 

[ehrenfest]

Would you care to show how the series diverges if |z| < 1?

I determined that if |z| > 1, then series converges. I did this by comparing |zⁿ| and |1 + zⁿ| and using the comparison test.

I highly doubt that would work because I used that comparison to show it diverges when |z| < 1. Please explain exactly how you compared the two series.

 

[ircdan]

I highly doubt that would work because I used that comparison to show it diverges when |z| < 1. Please explain exactly how you compared the two series.

why don't you show us what you did? it converges for |z| > 1

 

[e(ho0n3]

I highly doubt that would work because I used that comparison to show it diverges when |z| < 1. Please explain exactly how you compared the two series.

I will, once I see exactly how you showed that it diverges for |z| < 1.

 

[ehrenfest]

|\frac{1}{1+z^n}| \geq \frac{1}{1+|z|^n} \geq 1/|z|^n

So the sequence of terms doesn't even go to 0 when |z| < 1

 

[ircdan]

|\frac{1}{1+z^n}| \geq \frac{1}{1+|z|^n} \geq 1/|z|^n

So the sequence of terms doesn't even go to 0 when |z| < 1

this is wrong 1/(1 + |z|^n) >= 1/|z|^n

 

[ircdan]

for |z| > 1, |1/(1 + z^n)| <= |1/z^n| = |1/z|^n and |1/z| < 1, so we have convergence

 

[ircdan]

for |z| <= 1, suppose we had convergence, then lim n-> inf |1/(1 + z^n)| = 0, so lim 1/lim |1 + z^n| = 0, so lim |1 + z^n| = inf, but |1 + z^n| <= 1 + 1 = 2, a contradiction, so it cannot converge for |z| <= 1

 

[e(ho0n3]

I'm confused now. If Re(zⁿ) < 0, then |zⁿ| > |1 + zⁿ| right? If Re(zⁿ) >= 0, then |zⁿ| < |1 + zⁿ|. I think we have to consider both cases.

 

[ircdan]

I'm confused now. If Re(zⁿ) < 0, then |zⁿ| > |1 + zⁿ| right? If Re(zⁿ) >= 0, then |zⁿ| < |1 + zⁿ|. I think we have to consider both cases.

edit: you're right in that I made a mistake. Take n = 1, z = -1 + i, then |z| = sqrt(2) and |1 + z| = 1, so |z| > |1 + z|, i forgot these are complex numbers here

 

[e(ho0n3]

for |z| <= 1, suppose we had convergence, then lim n-> inf |1/(1 + z^n)| = 0, so lim 1/lim |1 + z^n| = 0, so lim |1 + z^n| = inf, but |1 + z^n| <= 1 + 1 = 2, a contradiction, so it cannot converge for |z| <= 1

By the triangle inequality, |1 + zⁿ| <= 1 + |z|ⁿ so

\frac{1}{1 + |z|^n} \le \frac{1}{|1 + z^n|} = \left|\frac{1}{1 + z^n}\right|

and furthermore

\sum_{n=0}^\infty \frac{1}{1 + |z|^n} \le \sum_{n=0}^\infty \left|\frac{1}{1 + z^n}\right|

So by the comparison test, if the LHS diverges, then so does the RHS. Now if |z| <= 1, then 1/2 <= 1/(1 + |z|ⁿ) <= 1 for all n and so the limit lies in [1/2, 1] and by the zero-test, the LHS diverges. Thus, so does the RHS.

The problem is this doesn't tell us if the original series of 1/(1 + zⁿ) diverges.