The sum of the series
is 44.
Step-by-step explanation:
The given series is ![\sum_{k=1}^{4}\left(2 k^{2}-4\right)=44](https://tex.z-dn.net/?f=%5Csum_%7Bk%3D1%7D%5E%7B4%7D%5Cleft%282%20k%5E%7B2%7D-4%5Cright%29%3D44)
To find the sum of the series, we need to substitute the values for k in the series.
![\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]](https://tex.z-dn.net/?f=%5Csum_%7Bk%3D1%7D%5E%7B4%7D%5Cleft%282%20k%5E%7B2%7D-4%5Cright%29%3D%5Cleft%5B2%281%29%5E%7B2%7D-4%5Cright%5D%2B%5Cleft%5B2%282%29%5E%7B2%7D-4%5Cright%5D%2B%5Cleft%5B2%283%29%5E%7B2%7D-4%5Cright%5D%2B%5Cleft%5B2%284%29%5E%7B2%7D-4%5Cright%5D)
Now, simplifying the square terms, we get,
![[2(1)-4]+[2(4)-4]+[2(9)-4]+[2(16)-4]](https://tex.z-dn.net/?f=%5B2%281%29-4%5D%2B%5B2%284%29-4%5D%2B%5B2%289%29-4%5D%2B%5B2%2816%29-4%5D)
Multiplying the terms,
![[2-4]+[8-4]+[18-4]+[32-4]](https://tex.z-dn.net/?f=%5B2-4%5D%2B%5B8-4%5D%2B%5B18-4%5D%2B%5B32-4%5D)
Subtracting the values within the bracket term, we get,
![-2+4+14+28](https://tex.z-dn.net/?f=-2%2B4%2B14%2B28)
Now, adding all the terms, we get the sum of the series,
![\sum_{k=1}^{4}\left(2 k^{2}-4\right)=44](https://tex.z-dn.net/?f=%5Csum_%7Bk%3D1%7D%5E%7B4%7D%5Cleft%282%20k%5E%7B2%7D-4%5Cright%29%3D44)
Thus, the sum of the series is ![\sum_{k=1}^{4}\left(2 k^{2}-4\right)=44](https://tex.z-dn.net/?f=%5Csum_%7Bk%3D1%7D%5E%7B4%7D%5Cleft%282%20k%5E%7B2%7D-4%5Cright%29%3D44)