Causality of z-transform $a^nu[n+1]$ To preface, this is not a homework related question but purely for self-study purposes.

I'm try to do the analyse of z-transform of $a^nu[n+1]$. It is clearly a non-causal signal, I try to explain it by using the definition.

Based on the causality of z-transform definition: If x(n) is a infinite duration causal sequence, ROC is exterior of the circle with radius a. i.e. $|z| &gt; |a|$.

I am not sure if my explanation is correct: 
$$\mathcal{Z}\big\{a^nu[n+1]\big\} = a^{-1}z+\sum_{n=0}^{\infty} (a/z)^{n}$$

The ROC of this function in z-plane is $|z| &gt; |a|$ except at $z = ∞$. 

It does not obey the causality definition because the boundary exists ( $|a|&lt;|z|&lt; ∞ $), i.e. the ROC pattern is a ring, it is a non-causal signal.

Am I correct? I am kinda confused with the causality definition.
You are correct. If the region of convergence of a right-sided signal (like the one you have) does not include infinity, then the signal is not causal.