For a non-zero complex number z , let (z) denote the principal argument of z , with -

Question

For a non-zero complex number z , let (z) denote the principal argument of z , with - < (z) . Let be the cube root of unity for which 0 < ( ) < . Let = ( _ n=1 ²⁰²⁵(- )^n ) . Then the value of 3 is ___________

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aligned & = (- + ^2- ^3+ +(- )²⁰²⁵ ) & = ( - ((- )²⁰²⁵-1 ) - -1 ) & = ( - - -1 (-2) ) & = ( -2 +1 ) & = ( -2 - ^2 ) & = ( 2 ) & = (2 ^2 ) & = -2 3 & 3 =-2 aligned

For a non-zero complex number z , let (z) denote the principal argument of z , with - < (z) . Let be the cube root of unity for which 0 < ( ) < . Let = ( _ n=1 ²⁰²⁵(- )^n ) . Then the value of 3 is ___________

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