Sums of powers

We consider the sums: \[S_N^{(k)}=\sum\limits_{n=0}^Nn^k\]

We can create a generating function: \[G_N(t)=\sum\limits_{k=0}^\infty\frac{t^k}{k!}S_N^{(k)}\]

Inserting the sum: \[G_N(t)=\sum\limits_{k=0}^\infty\frac{t^k}{k!}\sum\limits_{n=0}^Nn^k\]

We can reverse the sum order and combine the powers: \[G_N(t)=\sum\limits_{n=0}^N\sum\limits_{k=0}^\infty\frac{(tn)^k}{k!}\]

We recognize the infinite sum expression (which is why we chose the generating function form that we did): \[G_N(t)=\sum\limits_{n=0}^Ne^{tn}\]

But we also now recognize a geometric series: \[G_N(t)=\sum\limits_{n=0}^N(e^t)^n=\frac{e^{t(N+1)}-1}{e^t-1}\]