NumberForm[ReplaceAll[k{1,2,3,4}][((4*k-1)πI+2Log[2])(1+I)/4+(1+I)Sum[E^(-2kπn)/(2n!)(-2.0*I)^((I-1)n)FactorialPower[I*n-1,n-1],{n,1,40}]],]
Out[]//NumberForm=
{-2.0128+2.7031,-5.1512+5.8444,-8.2928+8.986,-11.434+12.128}
In[]:=
NumberForm[ReplaceAll[k{-1,-2,-3,-4}][((4k+1)πI+2Log[2])*(1-I)/4+(1-I)Sum[Exp[2kπn]/(2n!)(2.0I)^((-I-1)n)FactorialPower[-In-1,n-1],{n,1,40}]],40]
Out[]//NumberForm=
{-2.012775662931511-2.703074511590962,-5.151219459389824-5.844361280487999,-8.29280621812092-8.98595338867223,-11.43439886070235-12.1275460412436}
In[]:=
FindRoot[Exp[-z]-Sin[z],{z,Round[%6]},WorkingPrecision20]
Out[]=
FindRoot[Exp[-z]-Sin[z],{z,Round[%6]},WorkingPrecision20]
7
