$$\sum_{i=1}^n i^{-\alpha}=H_{n,\alpha}$$
the <A HREF="https://en.wikipedia.org/wiki/Harmonic_number#Generalized_harmonic_numbers">generalized Harmonic number.</A> For $\alpha>1$ one has the limit
$$\lim_{n\rightarrow\infty}H_{n,\alpha}=\zeta(\alpha),$$
the <A HREF="https://en.wikipedia.org/wiki/Riemann_zeta_function">Riemann zeta function.</A> The large-$n$ asymptotics is
$$H_{n,\alpha}=\zeta(\alpha)-\frac{1}{n^\alpha}\sum_{k=-1}^\infty\frac{B_{k+1}}{(k+1)!}\frac{(\alpha)_k}{n^k},$$
with $B_{k+1}$ Bernoulli numbers and $(\alpha)_k$ rising factorials. (See this <A HREF="https://math.stackexchange.com/questions/1405208/asymptotic-of-sum-r-1k-frac1r3-2-generalized-harmonic-number">MSE posting.)</A>