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 $x\in(0,\, 1)$.