I'm trying to prove that $ \sum_{k=1}^{\infty}\frac{A^k(z)}{k}=\ln\left(\frac{1}{1-A(z)}\right)$, (for those who know about the topic of analytical combinatorics, you may notice that it is the generating function of Cyc$(\mathcal{A})$). My proof attempt is the following:
$\displaystyle \sum_{k=1}^{\infty}\frac{A^k(z)}{k}=\sum_{k=0}^{\infty} \frac{A^{k+1}(z)}{k+1}=\sum_{k=0}^{\infty} \int_0^{A(z)} x^k\;\text{d}x=\int_0^{A(z)} \frac{\text{d}x}{1-x}=-\ln(1-A(z))=\ln\left(\frac{1}{1-A(z)}\right)$, where $A(z)$ is an exponential generating function associated with class $\mathcal{A}$.
It seems to me that the attempted proof is not bad, but my professor made an observation to me regarding $A(z)$ as the limit of integration. "Is $A(z)$ a number, or a function? And if it were a function, what sense would the integral have?" I honestly did not understand the meaning of the question, I consider that the integration limit should be considered as the generating function, but now I am a little doubtful.
And I would like to know where exactly my teacher's question is going, is there something that I am not understanding about equality or is there something that he is not understanding? I'm confused