Ratio Test

A series $\sum_{{n=1}}^{{\infty }}$an converges absolutely if $\sum_{{n=1}}^{{\infty }}$$\left\vert\vphantom{ a_{n}}\right.$an$\left.\vphantom{ a_{n}}\right\vert$ converges, in which case the series $\sum_{{n=1}}^{{\infty }}$an also converges. The ratio test states that a series $\sum_{{n=1}}^{{\infty }}$an converges absolutely (and therefore converges) if

$\displaystyle \lim_{{n\rightarrow \infty }}^{}$$\displaystyle \left\vert\vphantom{ \frac{a_{n+1}}{a_{n}}}\right.$$\displaystyle {\frac{{a_{n+1}}}{{a_{n}}}}$$\displaystyle \left.\vphantom{ \frac{a_{n+1}}{a_{n}}}\right\vert$ = L < 1


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

$\blacktriangleright$ Evaluate

$\lim\limits_{{n\rightarrow \infty }}^{}$${\dfrac{{a_{n+1}}}{{a_{n}}}}$ = ${\dfrac{{%
1}}{{2}}}$

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