How to recover the Harmonic numbers from this function?

As @Steven Clark suggested, let’s use a similar fact. Also note that the Hypergeometric function is just a Lerch Transcendent like here in your question.

$$F_2(1,n+1,n+2,x)=\,_2\text F_1(1,n+1;n+2;x)$$

Also try integrating $\sum\limits_{k=0}^{n-1} x^k$:

Therefore we have the following partial sums:

$$\int_0^1 \frac{1-x^n}{1-x}dx=\int_0^1\sum_{k=0}^{n-1}x^kdx=[-x^{n+1}Φ(x,1,n+1)-\ln(1-x)]_0^1=\sum_{k=0}^{n-1}\frac{1^{k+1}-0}{k+1}=\sum_{k=0}^{n-1}\frac1{k+1}=\text H_n$$

If you do not prefer the link:

$$Φ(x,1,n+1) =\sum_{k=0}^\infty \frac{x^k}{k+n+1},|x|<1$$ Please correct me and give me feedback!