Integral InfiniteSeries SMT Challenging

Problem - 4587

Given that the value $\ln(2)$ is not the root of any polynomial with rational coefficients. For any nonnegative integer $n$, let $p_n(x)$ be the unique polynomial with integer coefficients such that $$p_n(\ln(2)) =\int_1^2 (ln(x))^n dx$$

Compute the value of the $$\sum_{n=0}^{\infty}\frac{1}{p_n(0)}$$

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