Given the Vandermonde matrix:
$\begin{pmatrix}1^0 & 1^1 & 1^2 & ... & 1^n \\ 2^0 & 2^1 & 2^2 & ... & 2^n \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ N^0 & N^1 & N^2 & ... & N^n\end{pmatrix}$
the row sums can be calculated using the closed-form expression for a geometric series (or the column sums calculated using the Bernoulli expression for n-th powers). Is there a closed-form expression to simplify the sum of all elements within the Vandermonde matrix? That is, is there a closed-form expression for:
$\begin{align*} 1+n+\sum_{r=2}^{N}\frac{(1-r^{n+1})}{(1-r)} \end{align*}$
Or alternatively, is there any way to expedite summing all elements within the Vandermonde matrix? Thanks.