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$$\sum_{i=1}^n i^{-\alpha}=H_{n,\alpha}$$ the generalized Harmonic number. For $\alpha>1$ one has the limit $$\lim_{n\rightarrow\infty}H_{n,\alpha}=\zeta(\alpha),$$ the Riemann zeta function. 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 MSE posting.)