Root Test

The root test states that a series $\sum_{{n=1}}^{{\infty }}$an converges absolutely (and therefore converges) if

$\displaystyle \lim_{{n\rightarrow \infty }}^{}$$\displaystyle \sqrt[n]{{\left\vert a_{n}\right\vert }}$ = L < 1

To verify convergence of $\sum_{{n=1}}^{{\infty }%
}$${\frac{{n^{2}}}{{2^{n}}}}$ using the root test, note the following.

$\blacktriangleright$ Evaluate

$\lim\limits_{{n\rightarrow \infty }}^{}$$\sqrt[n]{{\left\vert a_{n}\right\vert }%
}$ = ${\dfrac{{1}}{{2}}}$

Thus, L = 1/2, which is less than 1, again showing that the series converges absolutely.