Ratio Test Series: ((k+1)^2)/(k^2) * (2^k)/(2^k+1)

[shivers20]

I am having a hard time on multiplying these two fractions for Ratio Test. How did they get that answer. How do I multiply out when there are variables presnt such as k? This simple multiplication step is holding me up. Thanks in advance.

((k+1)^2)/(k^2) * (2^k)/(2^k+1) = ((k+1)/(k))^2 * (1/2)

 

[pka]

\(\displaystyle \L
\begin{array}{l}
\sum {\frac{{n^2 }}{{2^n }}} \\
\frac{{a_{n + 1} }}{{a_n }} = \frac{{\frac{{\left( {n + 1} \right)^2 }}{{2^{n + 1} }}}}{{\frac{{n^2 }}{{2^n }}}} = \frac{{\left( {n + 1} \right)^2 }}{{n^2 }}\left( {\frac{{2^n }}{{2^{n + 1} }}} \right) = \left( {\frac{{n + 1}}{n}} \right)^2 \left( {\frac{1}{2}} \right) \\
\end{array}\)

Recall that
\(\displaystyle \L
\frac{{a^k }}{{b^k }} = \left( {\frac{a}{b}} \right)^k\)