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I came cross the following serie :

$$\sum\limits_{\mathbf k \in \mathbb N^d} e^{\langle \mathbf r, \mathbf k \rangle}$$

What would be the conditions on the d-dimensional real vector $\mathbf r$ for the convergence of this serie ? What would be the obtained value of the serie in this case ?

More generally, is there any litterature / a name / some referenc ei could read about this sort of multivariate power serie ?

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    $\begingroup$ Fubini-Tonelli's theorem tells you that this sum equals (wether it is finite or infinite) the product of the sums $\sum_{k_i\in\mathbf{N}}e^{r_i k_i}$. A necessary and sufficient condition for convergence is therefore that all the coordinates of your vector $r$ to be negative, and in that case the sum equals $\prod_{i=1}^d \frac{1}{1-e^{r_i}}$ $\endgroup$ Nov 26, 2020 at 16:23
  • $\begingroup$ Yes, indeed, this simplify the problem. Thanks a lot, this is exactly where i was stuck ! Edit: If you post this as an answer, i'll accept it. $\endgroup$
    – lrnv
    Nov 26, 2020 at 16:29

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