Let 0<a<1 and consider the series
a^2 + a + a^4 + a^3 + ... +a^(2n) + a^(2n-1) + ...
Show that the Root test applies but the ratio test does not apply.
Thanks in advance
Is there any reason we can't write this as $\displaystyle \displaystyle \begin{align*} a + a^2 + a^3 + a^4 + \dots \end{align*}$ and state that it is a geometric series with $\displaystyle \displaystyle \begin{align*} t_1 = a \end{align*}$ and $\displaystyle \displaystyle \begin{align*} r = a \end{align*}$, therefore the series will be convergent if $\displaystyle \displaystyle \begin{align*} |a| < 1 \end{align*}$?