The principal argument of z is 0, which corresponds to choice (a) π/2 .
The expression F(z)=1+i+ 2/ 1+i +⋯+ 1/n −i involves a sequence of complex numbers being added together.
Let's simplify the expression for F(z) step by step:
F(z)=1+i+ 2/ 1+i +⋯+ 1/n −i
To simplify 2/ 1+i , multiply the numerator and denominator by the conjugate of the denominator to eliminate the complex number from the denominator:
2/ 1+i = 2(1−i)/ (1+i)(1−i) = 2−2i/ 1+1 =1−i
So, the expression becomes:
F(z)=1+i+(1−i)+⋯+ 1/n −i
Simplifying further:
F(z)=2+ 1/n
Now, let's analyze the expression F(z):
F(z)=2+ 1/n
As n approaches infinity, 1/n approaches zero, and F(z) approaches 2.
The complex number z represented by this expression must lie on the line passing through the origin and the point 2 in the complex plane. The argument of z can be found as the angle made by this line with the positive real axis.
The argument of z can be calculated as:
Argument of z=arctan( Re(z)/Im(z) )
For the point 2 on the complex plane, the real part is 2 and the imaginary part is 0. So, the argument of z is:
Argument of z=arctan( 0/2 )=arctan(0)=0
Therefore, the principal argument of z is 0, which corresponds to choice (a) π/2 .