For which natural number $k$ does the expression $\frac{k^{2}}{1.001^{k}}$ attain its maximum value?

## List 9
We are looking for the value of $k$ for which the term of the sequence is maximum:

$$
\frac{1^{2}}{1,001}, \frac{2^{2}}{1,001^{2}}, \frac{3^{2}}{1,001^{3}}, \ldots, \frac{k^{2}}{1,001^{k}}, \ldots
$$

Consider the following equivalent inequalities:

$$
\frac{(k+1)^{2}}{1,001^{k+1}}1000 \Leftrightarrow k>2000
$$

Thus, the sequence decreases strictly for $k \geq 2001$ and increases strictly for $k \leq 2000$. Therefore, the largest term of the sequence corresponds to $k=2001$.

## List 9

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