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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.

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Your expression doesn't make sense for $r=1$. – Robert Israel Oct 29 '12 at 0:14
Thank you. I have corrected the expression to begin with r = 2. – SPSmith Oct 29 '12 at 0:23
Do you need an exact solution or asymptotic? The latter is easy to obtain by approximating sums with integrals – Alex Oct 29 '12 at 0:53
An exact solution. Thanks. – SPSmith Oct 29 '12 at 0:56
Did you notice $n = N-1$? – cactus314 Oct 29 '12 at 21:22

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