(Pi/2 - E I) - 
    NSum[(I)^n *((-n)^(1/n) - 1), {n, 1, Infinity}, 
    WorkingPrecision -> 30, Method -> "AlternatingSigns"] + (
    3^(3/4) Sqrt[19 - 5^(2/3)])/10^3 + Im[(23 + 9*I)^(3/19)/10] I

= -1.956013164857547142  10^-10 + 5.429398424534603227  10^-10 I

That is .

    Pi - Abs[NSum[I^n*((-n)^(1/n) - 1), {n, 1, Infinity}, 
       WorkingPrecision -> 30, Method -> "AlternatingSigns"]] - 
       N[((1/100)*(19822603/2918343)^(1/3)*
          Pi + (1/100)*(3782797/8082127)^(2/3)*Pi^2)/20, 30]
=1.184984684*10^-19

That is .