I'm going through the proof from the book 'Bernoulli Numbers and Zeta Functions' and there's a step that I don't understand. My best guess at the minute is Taylor series. (It's the line highlighted in yellow in the image) Thanks
Similarly, taking the derivatives successively, we obtain $$ S_{k}^{(j)}(0)=k(k-1) \cdots(k-j+2) b_{k-j+1} \quad(2 \leq j \leq k+1) $$ Finally, we have $$ \begin{aligned} \color{red}{S_{k}(x)} &=\sum_{j=0}^{k+1} \frac{S_{k}^{(j)}(0)}{j !} \color{red}{x}^{j} \\ &=\sum_{j=1}^{k+1} \frac{1}{k+1}\left(\begin{array}{c} k+1 \\ j \end{array}\right) b_{k-j+1} x^{j} \quad\left(S_{k}^{(0)}(0)=0\right) \\ &=\frac{1}{k+1} \sum_{j=0}^{k}\left(\begin{array}{c} k+1 \\ j \end{array}\right) b_{j} x^{k+1-j} \end{aligned} $$
I suppose $S_k(n)=\sum_{i=1}^n i^k$, which is a polynomial in $n$ of degree $k+1$. This is just Taylor's formula applied to this polynomial (remeber Taylor's formula at order $k+1$ is a exact formula for polynomials of degree $\le k+1$.