sequences and series – Show that if $|z|


Show that if $|z| < 1$, then
begin{align*}
frac{z}{1+z} + frac{2z^{2}}{1+z^{2}} + frac{4z^{4}}{1+z^{4}} + frac{8z^{8}}{1+z^{8}} + ldots
end{align*}

converges

MY ATTEMPT

Let us start by noticing that
begin{align*}
frac{z}{1-z} – frac{z}{1-z} + frac{z}{1+z} = frac{z}{1-z} + frac{z – z^{2} -z – z^{2}}{1-z^{2}} = frac{z}{1-z} – frac{2z^{2}}{1-z^{2}}
end{align*}

Similarly, we have that
begin{align*}
frac{z}{1-z} – frac{2z^{2}}{1-z^{2}} + frac{2z^{2}}{1+z^{2}} = frac{z}{1-z} + frac{2z^{2} – 2z^{4} – 2z^{2} – 2z^{4}}{1-z^{4}} = frac{z}{1-z} – frac{4z^{4}}{1-z^{4}}
end{align*}

Based on this pattern, it can be proven by induction the given series is the same as
begin{align*}
frac{z}{1-z} = z + z^{2} + z^{3} + ldots = sum_{n=1}^{infty}z^{n}
end{align*}

which converges, since it is the geometric series inside the circle $|z| < 1$ (we may apply the ratio test, for instance).

Could someone tell me if my approach is correct? Any contribution is appreciated.