I have an issue with the following problem involving iterated integrals:
Context.
Let $(\omega_1, \dots, \omega_n)$ be complex-valued differential 1-forms defined in a real interval $[a,b]$. For all $i \in \{1, \dots, n\}$, we have $\omega_i = f_i(s)ds$, where $f_i$ is a complex function.
Definition. We define the iterated integral $\int_a^b \omega_1 \dots \omega_n$ inductively by \begin{equation*} \begin{cases} \int_a^b \omega_1 = \int_a^b f_1(s) ds \\ \int_a^b \omega_1 \dots \omega_n = \int_a^b f_1(s) \left(\int_a^s \omega_2 \dots \omega_n\right) ds \end{cases} \end{equation*}
Now, consider the two differential 1-forms \begin{equation*} \Omega_0 = \frac{ds}{s} \text{ and } \Omega_1 = \frac{ds}{s-1} \end{equation*}
I am struggling in the demonstration of the following proposition :
Proposition. Let $k \in \mathbb{N}^*$, and $p_1, q_1, \dots, p_k, q_k \in \mathbb{N}$ (such that $q_1, p_2, q_2, \dots p_{k-1}, q_{k-1}, p_k \geq 1$). The iterated integral $$ \int_{\varepsilon}^{1 - \varepsilon} \Omega_0^{p_1}\Omega_1^{q_1} \dots \Omega_0^{p_k}\Omega_1^{q_k} $$ converges when $\varepsilon$ tends to $0$ if and only if $p_1 \geq 1$ and $q_k \geq 1$.
Examples where it works
By induction on $p \in \mathbb{N}*$, we prove that $$ \int_{\varepsilon}^{1 - \varepsilon} \Omega_0^{p} = \frac{1}{p!} \left(\ln\left(\frac{1-\varepsilon}{\varepsilon}\right)\right)^p $$ It works since $q_k = q_1 = 0$ and we have the divergence.
The same thing holds with $$ \int_{\varepsilon}^{1 - \varepsilon} \Omega_1^{q} = \frac{1}{q!} \left(\ln\left(\frac{\varepsilon}{1 - \varepsilon}\right)\right)^q $$ since $p_1 = 0$.
Last example, I tried $$\int_{\varepsilon}^{1 - \varepsilon} \Omega_0 \Omega_1$$ converges to $-\zeta(2)$ (where $\zeta$ denotes Riemann zeta function) when $\varepsilon$ tends to $0$. Here $k = 1, p_1 = 1$ and $q_1 = 1$.
We remark that we can get the conclusion by doing the full computation of the iterated integral. This is impossible in a more general context like in the proposition. Is there any way to use classical convergence tools (like comparaison, equivalents, etc) to prove the proposition ?
Thanks in advance for any help or hint.
Best regards.
K. Y.