Looking for a method to maximize a sum

Question

I need a method to find a sequence $(n_k)_k$ of positive real numbers that maximizes the following sum: $$ \sum^{+\infty}_{k=0}(a_k\, n_k-n_k^2\,e^{-r\,k}), $$ where $(a_k)_k$ is a given sequence and $r$ is a data.

Answer 1

The stationary points are computed as

$$ \nabla\sum_{k=0}^{\infty}(a_kn_k-n^2_ke^{-r k}) = 0 $$

giving

$$ a_k - 2n_k e^{-r k} = 0, \ \ k = 0,1,\cdots $$

or

$$ n_k = \frac{1}{2}a_k e^{r k} $$

giving

$$ \sum_{k=0}^{\infty}(a_kn_k-n^2_ke^{-r k}) = \frac{1}{2}\sum_{k=0}^{\infty}a_k^2e^{r k} $$