By the Euler-Maclaurin formula,
$$\sum_{k=1}^nk^i=\zeta(-i)+\frac{n^{i+1}}{i+1}+\frac{n^i}2+\frac{in^{i-1}}{12}-\frac{i(i-1)(i-2)n^{i-3}}{720}+\mathcal O(n^{i-5})$$
this is not an exact formula, but rather an approximation to the problem, where $\zeta(-i)\approx0.0033002237+0.4181554491i$ is the Riemann zeta function. A few values are given to show how it approximates:
$$\begin{array}{c|c}n&\sum&\zeta\\\hline1&1&0.49913+0.00010i\\2&1.76924+0.63896i&1.38459+0.31946i\\3&2.22407+1.52954i&1.99665+1.08425i\end{array}$$
Ok, not so good for small values, but it gets better.
$$\begin{array}{c|c}n&\sum&\zeta\\\hline10&0.41898+7.84548i&0.37600+7.47349i\\20&-8.93237+11.8340i&-8.43768+11.76128i\\30&-18.82708+10.93402i&-18.34384+11.06237i\end{array}$$
As you can see, the accuracy increases as $n\to\infty$.
Alternative forms include
$$\sum_{k=1}^nk^i=\zeta(-i)-\zeta(-i,n+1)=H_n^{(-i)}$$
where $\zeta(s,q)$ is the Hurwitz zeta function and $H_n^{(p)}$ is the generalized harmonic number.