Does there exist a degree-2 polynomial with positive acceleration such that the real extension of the harmonic numbers surpasses it for all future values?
This was too big of a title and it's a really big mouthful, so I'll further elaborate on what I'm looking for.
$$ p(x) = ax^2 + bx + c \quad \quad a > 0, \quad (b, c) \in \mathbb R^2 $$
Above is the family of polynomials I care about. Positive acceleration (the highest degree term's coefficient), other coefficients just need to be real.
$$ H_k = \sum_{n = 1}^x \frac 1 n $$
Above is the definition of the $k$th Harmonic Number.
$$ H_x = H_k \quad \quad x \in \mathbb N $$
$H_x$ will be the real extension of $H_k$ to the reals.
$$ H_x = \int_0^1 \frac{1 - y^x}{1 - y} dy $$
My question is now, does there exist $p(x)$ as defined above, such that $p(x_0 + c) < H_{x_0 + c}$ for some $x_0$ and $c > 0$?