_ n=1 ⁴ (-1)^ 2n i^ 2n =

Step-by-step solution

The given expression is _ n=1 ⁴ (-1)^ 2n i^ 2n Since (-1)^ 2n = ((-1)^2)^n = 1^n = 1 for all integers n , and i^ 2n = (i^2)^n = (-1)^n , the expression simplifies to: _ n=1 ⁴ 1 (-1)^n = _ n=1 ⁴ (-1)^n Expanding the sum for n = 1, 2, 3, 4 : (-1)^1 + (-1)^2 + (-1)^3 + (-1)^4 -1 + 1 - 1 + 1 = 0 Answer: 0