The discussion revolves around the convergence of the series $\displaystyle\sum_{n=0}^{\infty}\left(\frac{z}{z+1}\right)^n$ for complex values of $z$. Participants explore the application of the ratio test and the conditions under which the series converges.
Participants do not reach a consensus on the correct approach to solving the inequality for convergence. There are multiple interpretations and some confusion regarding the mathematical steps involved.
Participants express uncertainty about the transformations used in the inequalities and the correct application of the modulus of complex numbers. There are unresolved questions about the steps taken in the mathematical reasoning.
You solve that inequality..surely you can do that, no?dwsmith said:$\displaystyle\sum_{n=0}^{\infty}\left(\frac{z}{z+1}\right)^n$$z\in\mathbb{C}$
By the ratio test,
$\displaystyle\left|\frac{z}{z+1}\right|<1$
I am stuck at this part. How do I find the z such that some is convergence?
AlexYoucis said:You solve that inequality..surely you can do that, no?
dwsmith said:Apparently I can't because I keep getting it wrong.
AlexYoucis said:Ok, so we need to solve $|z|<|z+1|$ or $x^2+y^2<(x+1)^2+y^2$ or $x^2<(x+1)^2$ or $x>\frac{-1}{2}$.
dwsmith said:How did you go from this $|z|<|z+1|$ to this $x^2+y^2<(x+1)^2+y^2$??
AlexYoucis said:Let $z=x+iy$.
dwsmith said:I did but I don't see what happened to all the i's.
AlexYoucis said:If $z=x+iy$ then $|z|=x^2+y^2$.
dwsmith said:Shouldn't it be square rooted?