B= -1^2 + 2^2 - 3^2 + 4^2 -...+ (-1)^n . n^2

The given expression can be written as:

B = (-1)^2 + 2^2 - 3^2 + 4^2 - ... + (-1)^n . n^2

Let's break it down into two separate series:

Series 1: (-1)^2 - 3^2 + 5^2 - ...
Series 2: 2^2 + 4^2 + 6^2 + ...

Series 1 is an alternating series with a common difference of 2 between each term. The general term of this series can be written as:

a(n) = (-1)^n . (2n - 1)^2

To find the sum of this series, we can use the formula for the sum of an alternating series:

S1 = (n/2) . [a(1) + a(n)]
= (n/2) . [(-1)^1 . (2(1) - 1)^2 + (-1)^n . (2n - 1)^2]
= (n/2) . [1 + (-1)^n . (2n - 1)^2]

Series 2 is a regular series with a common difference of 2 between each term. The general term of this series can be written as:

a(n) = (2n)^2

To find the sum of this series, we can use the formula for the sum of a regular series:

S2 = (n/2) . [a(1) + a(n)]
= (n/2) . [(2(1))^2 + (2n)^2]
= (n/2) . [4 + 4n^2]
= 2n + 2n^3

Now, let's find the sum of the original series B by subtracting Series 1 from Series 2:

B = S2 - S1
= 2n + 2n^3 - (n/2) . [1 + (-1)^n . (2n - 1)^2]
= 2n + 2n^3 - (n/2) - (-1)^n . (n/2) . (2n - 1)^2

Therefore, the sum of the series B is given by the expression:

B = 2n + 2n^3 - (n/2) - (-1)^n . (n/2) . (2n - 1)^2