$$\sum_{i=1}^n i^{-\alpha}=H_{n,\alpha}$$
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>