Root test of $\frac{1}{2n}$

Question

Let $a_n=\frac{1}{2n}$

Then $|a_n|^\frac{1}{n}=\frac{1}{2n}^\frac{1}{n}$ And $\frac{1}{2n}^\frac{1}{n}<1 $ for every $n\geqslant1$.

So $L=\limsup_{n\rightarrow\infty} |a_n|^\frac{1}{n}<1$

By root test, $\sum_{n=0}^\infty a_n$ converges.

But as we know, $\sum_{n=0}^\infty a_n$ diverges by p-series test.

What did I missed?

Answer 1

$|a_n|^{\frac 1 n} <1$ for all $n$ does not imply that $\lim |a_n|^{\frac 1 n} <1$. It only implies $\lim |a_n|^{\frac 1 n} \leq 1$. In this case equality holds, so root test cannot be used.

Answer 2

We have $L=1$ and so the root test is inconclusive.