This question is motived by the following observation: Consider the principal value integral $$ P.V.\int_{-\eta_1}^{\eta_2}\frac{1}{a_1 x + a_2 x^2 + a_3 x^3 + \cdots} $$
Here I choose $\eta_1$ and $\eta_2$ small enough to only contain the singular point at x=0. If $a_1\neq0$, the denominator behaves like $a_1 x$ when it's sufficiently close to $0$ and so the principal value integral gives finite value. So, the principal value integral in this case is always converge. However, when $a_1=0$, principal value integral can be diverge if $a_2\neq0$. So, this means that for this formula a necessary condition for principal value integral to diverge is when $a_1=0$.
Now if we have a general form: $$ P.V.\int_{-\eta_1}^{\eta_2}\frac{1}{f(x)} $$ where $f(0)=0$ and in $C^{\infty}$, we can always Tayler expand $f(x)$. In this case, looks for me that the necessary condition for principal value to diverge is $f'(0)=0$. Do you think this is correct and does anyone know some theorems related to this? Thank you!