How to prove $5 + \sum_{n=1}^{\infty} \sqrt[n] 5$ is divergent? As the title describes, I tried ratio test and root test, but the answer is 1 for both.

You may observe that
$$5^{\frac{1}{n} }>1$$ for any positive integer $n$ and consequently $$ \sum_{n=1}^{k} 5^{\frac{1}{n} } > k$$ and consider the $\lim_{k\to \infty}$ to conclude that the series is divergent.
