Cauchy's criterion for series convergence is that for a $\sum\_{n=1}^{\infty} {a\_n}$ if

$$\limsup\sqrt[n]{a\_n} < 1$$

then the series converges. If its $>1$ then it diverges and if its $1$ then this criterion doesnt give an answer

If we have

$$\sum\_{n=1}^{\infty}\frac{1}{n^2}$$

then how come that this criterion gives us an answer?

Because $\lim\sqrt[n]{a} = 1$ if $a>0$

But textbook says that the above series converges. So maybe the above limit is actually less than $1$? Or have i misinterpreted Cauchy's criterion?