To find the sum of the series [tex]\(\sum_{k=1}^3 \left(k^2 - 3\right)\)[/tex], we need to evaluate the expression for each value of [tex]\(k\)[/tex] from 1 to 3, and then sum the results.
Let's break down the steps:
1. Evaluate the expression [tex]\(k^2 - 3\)[/tex] for [tex]\(k = 1\)[/tex]:
[tex]\[
(1)^2 - 3 = 1 - 3 = -2
\][/tex]
2. Evaluate the expression [tex]\(k^2 - 3\)[/tex] for [tex]\(k = 2\)[/tex]:
[tex]\[
(2)^2 - 3 = 4 - 3 = 1
\][/tex]
3. Evaluate the expression [tex]\(k^2 - 3\)[/tex] for [tex]\(k = 3\)[/tex]:
[tex]\[
(3)^2 - 3 = 9 - 3 = 6
\][/tex]
So, the terms of the series are:
[tex]\[
-2, 1, 6
\][/tex]
4. Sum these terms:
[tex]\[
-2 + 1 + 6 = -1 + 6 = 5
\][/tex]
Thus, the sum of the series [tex]\(\sum_{k=1}^3 \left(k^2 - 3\right)\)[/tex] is [tex]\(\boxed{5}\)[/tex].
