A probelem of Mathematical Physics which 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 $k\in\mathbb{N}$. Is there a way to compute the analytical expression of the limit

$$ \lim_{k\to n} I_k $$ where $n$ is an integer number equal or greater than $2$?

Is there a way to compute, at least, the limit $\lim_{k\to2}$ of $I_k$?

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