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 :)

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- Nov 15th 2012, 07:51 PMAainaaRozTests for nonabsolute convergence
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 :) - Nov 15th 2012, 08:13 PMProve ItRe: Tests for nonabsolute convergence
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 $\displaystyle \displaystyle \begin{align*} |a| < 1 \end{align*}$?

- Nov 15th 2012, 09:20 PMAainaaRozRe: Tests for nonabsolute convergence
I don't think so. Because the question specifically asks to show that root test applies but ratio test doesn't.