Answer
$\sum\limits_{k=2}^{n}{{{\left( -1 \right)}^{k}}{{k}^{2}}}.$
Work Step by Step
$4-9+16-25+\ldots +{{\left( -1 \right)}^{n}}{{n}^{2}}$
This is the sum of the square of the natural numbers, and the sum alternates between positive and negative.
The value of $k$ varies from $k=2$ to $k=n$.
Thus, the sigma notation is,
$\sum\limits_{k=2}^{n}{{{\left( -1 \right)}^{k}}{{k}^{2}}}$
Thus, the sigma notation for the sum$4-9+16-25+\ldots +{{\left( -1 \right)}^{n}}{{n}^{2}}$ is $\sum\limits_{k=2}^{n}{{{\left( -1 \right)}^{k}}{{k}^{2}}\text{ }}$.
