A problem of Mathematical Physics that I am working on involves the computation of a certain integral. Part of the result reads:

$$ I_k:= [\beta_x( -1 - k, 0) + H_{-2 - k}]x^k $$

where $\beta_x( -1 - k, 0)$ is the incomplete Beta function, with $x\in(0, \, 1)$, and $H_{-2-k}$ is an harmonic number.

From physical considerations, I expect this integral to be well defined for $0<x<1$. Of special importance are the cases of $k\ge 2$ and $k\in\mathbb{N}$.

Question. Is there a way to compute the analytical expression of the limit $$ \lim_{k\to n} I_k$$ where $n$ is an integer number $\geq 2$? Is there a way to compute, at least, $\lim_{k\to2} I_k$?

I hope that $I_n$ can be expressed as a function / special function / series of the argument $x$ in the domain $x\in(0,\, 1)$.