how to calculate the sum of $1^k+2^k+3^k+\dots+n^k$

Question

how to calculate this？I have some indistinct memory that it can be solved by induction, but I forget how to do it.

Answer 1

Hint. Call this sum $S_k$.

Develop $(i+1)^{k+1}-i^{k+1}$ by the binomial formula and add these for $i=1$ to $n$. The left-hand side will give you a telescoping sum and the right-hand side will be a linear combination of $S_1, S_2,\dots, S_k$, whence a recursive relation between all these sums.