Thread: Tests 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

Is there any reason we can't write this as and state that it is a geometric series with and , therefore the series will be convergent if ?

I don't think so. Because the question specifically asks to show that root test applies but ratio test doesn't.