<s>[INST] Graph the set of all complex numbers $z_n=\frac{2n+1}{n-i},n\in\mathbb R$ I'm a bit rusty on my complex numbers, how would you solve the following problem on paper?

Determine and sketch (graph) the set of all complex numbers of form:
  $$z_n=\frac{2n+1}{n-i},n\in\mathbb R$$

Rationalizing yields $$S=\left\{z_n\in\mathbb C : z_n=\frac{n + 2 n^2}{1 + n^2}+\frac{1 + 2 n}{1 + n^2}i\right\}$$
How do I proceed to sketch (graph) this now on paper? (Wolframalpha yields a circle)
I assume I need to find the center and the radius of that circle which would be enough to sketch the graph, but I can't quite proceed from this point on. 
 [/INST] I'll use $x$ instead of $n$.
Note that$$\frac{2x^2+x}{x^2+1}=1+\frac{x^2+x-1}{x^2+1}=1+\frac{2x^2+2x-2}{2(x^2+1)}$$and that$$\frac{2x+1}{x^2+1}=\frac12+\frac{-x^2+4x+1}{2(x^2+1)}.$$So$$\frac{2x+1}{x-i}=1+\frac i2+\frac{2x^2+2x-2}{2(x^2+1)}+\frac{-x^2+4x+1}{2(x^2+1)}i.$$Now, observe that$$\left(\frac{2x^2+2x-2}{2(x^2+1)}\right)^2+\left(\frac{-x^2+4x+1}{2(x^2+1)}\right)^2=\frac54.$$Therefore, your set is contained in the circle with center $1+\frac i2$ and radius $\frac{\sqrt5}2$.
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