_ k=1 ²⁰²⁶ ⁻¹ ( k 4 ) =

Step-by-step solution

Let f(k) = ⁻¹ ( k 4 ) . The function k 4 is periodic with a period of 8 . Let us evaluate the sum of f(k) over one complete period, from k=1 to k=8 : f(1) = ⁻¹ ( 4 ) = ⁻¹ ( 1 2 ) = 4 f(2) = ⁻¹ ( 2 4 ) = ⁻¹(0) = 0 f(3) = ⁻¹ ( 3 4 ) = ⁻¹ (- 1 2 ) = - 4 f(4) = ⁻¹ ( 4 4 ) = ⁻¹(-1) = - 2 f(5) = ⁻¹ ( 5 4 ) = ⁻¹ (- 1 2 ) = - 4 f(6) = ⁻¹ ( 6 4 ) = ⁻¹(0) = 0 f(7) = ⁻¹ ( 7 4 ) = ⁻¹ ( 1 2 ) = 4 f(8) = ⁻¹ ( 8 4 ) = ⁻¹(1) = 2 Summing these values, we get: _ k=1 ⁸ f(k) = 4 + 0 - 4 - 2 - 4 + 0 + 4 + 2 = 0 Since the sum over any 8