To find the sum of the series [tex]\(\sum_{k=1}^4 (2k^2 - 4)\)[/tex], we will evaluate the expression term by term and then sum the results.
Let's break it down step by step:
1. Evaluate the expression for [tex]\( k = 1 \)[/tex]:
[tex]\[
2(1)^2 - 4 = 2 \cdot 1 - 4 = 2 - 4 = -2
\][/tex]
2. Evaluate the expression for [tex]\( k = 2 \)[/tex]:
[tex]\[
2(2)^2 - 4 = 2 \cdot 4 - 4 = 8 - 4 = 4
\][/tex]
3. Evaluate the expression for [tex]\( k = 3 \)[/tex]:
[tex]\[
2(3)^2 - 4 = 2 \cdot 9 - 4 = 18 - 4 = 14
\][/tex]
4. Evaluate the expression for [tex]\( k = 4 \)[/tex]:
[tex]\[
2(4)^2 - 4 = 2 \cdot 16 - 4 = 32 - 4 = 28
\][/tex]
Next, we sum these individual results:
[tex]\[
-2 + 4 + 14 + 28
\][/tex]
Now, add these values together:
[tex]\[
-2 + 4 = 2
\][/tex]
[tex]\[
2 + 14 = 16
\][/tex]
[tex]\[
16 + 28 = 44
\][/tex]
Therefore, the sum of the series [tex]\(\sum_{k=1}^4 (2k^2 - 4)\)[/tex] is
[tex]\[
\boxed{44}
\][/tex]
