The sum of the series [tex]\sum_{k=1}^{4}\left(2 k^{2}-4\right)[/tex] is 44.
Step-by-step explanation:
The given series is [tex]\sum_{k=1}^{4}\left(2 k^{2}-4\right)=44[/tex]
To find the sum of the series, we need to substitute the values for k in the series.
[tex]\sum_{k=1}^{4}\left(2 k^{2}-4\right)=\left[2(1)^{2}-4\right]+\left[2(2)^{2}-4\right]+\left[2(3)^{2}-4\right]+\left[2(4)^{2}-4\right][/tex]
Now, simplifying the square terms, we get,
[tex][2(1)-4]+[2(4)-4]+[2(9)-4]+[2(16)-4][/tex]
Multiplying the terms,
[tex][2-4]+[8-4]+[18-4]+[32-4][/tex]
Subtracting the values within the bracket term, we get,
[tex]-2+4+14+28[/tex]
Now, adding all the terms, we get the sum of the series,
[tex]\sum_{k=1}^{4}\left(2 k^{2}-4\right)=44[/tex]
Thus, the sum of the series is [tex]\sum_{k=1}^{4}\left(2 k^{2}-4\right)=44[/tex]
