root test

Root test

Let ${\sum }_{k=1}^{\infty }$ be a series.
i) if there exists some $q<1$ such that

$$\sqrt[k]{|a_k|}\le q\text{for almost all }k,$$

then ${\sum }_{k=1}^{\infty }{a}_k$ is absolutely convergent.
ii) Converseely, if

$$\sqrt[k]{|a_k|}\ge 1\text{for infinitely many }k,$$

then ${\sum }_{k=1}^{\infty }{a}_k$ is divergent.

In other words: If the $k$-th root of the absolute value of the terms of a series is eventually smaller than some $q<1$, then the series is absolutely convergent. If the $k$-th root of the absolute value of the terms of a series is greater than or equal to $1$ infinitely many times, then the series is divergent.

Proof from page 132

And writing it with lim sup, plus examples etc. pp