I used the root test for the series $\displaystyle \sum_{n=1}^{\infty} \left(\frac{cosn}{2}\right)^n$. I showed that $\displaystyle 0\leq \lvert \frac{cosn}{2}\rvert \leq \frac{1}{2}$. So, $\displaystyle \lim_{n\to\infty}\lvert\frac{cosn}{2}\rvert\leq \frac{1}{2}<1$. By the root test, the series converges absolutely. My professor told me that the flaw here is that the limit above does not exist. I agree the limit does not exist because $\displaystyle \lvert\frac{cosn}{2}\rvert$ oscillates between $\displaystyle 0$ and $\displaystyle \frac{1}{2}$. However, I fail to see why my argument does not work here. She suggested that I use the comparison test and compare the series with $\displaystyle \sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^n$. By the comparison test, the original series converges absolutely. Is it a coincidence that the "pseudo" root test I used yielded the same answer as the comparison test? Can we say that if $\displaystyle \lvert a_n\rvert^{\frac{1}{n}}<1$, then $\displaystyle \sum_{n=1}^{\infty} a_n$ converges absolutely?
I appreciate any help on this.