Đáp án:
$\dfrac{{p\left( {1 - q} \right)}}{{q\left( {1 - p} \right)}}$
Giải thích các bước giải:
$\begin{array}{l}
p + {p^2} + ... + {p^n} = \dfrac{{p\left( {1 - {p^n}} \right)}}{{1 - p}}\\
q + {q^2} + ... + {q^n} = \dfrac{{q\left( {1 - {q^n}} \right)}}{{1 - q}}\\
\Rightarrow \lim \dfrac{{p + {p^2} + ... + {p^n}}}{{q + {q^2} + ... + {q^n}}} = \lim \dfrac{{\dfrac{{p\left( {1 - {p^n}} \right)}}{{1 - p}}}}{{\dfrac{{q\left( {1 - {q^n}} \right)}}{{1 - q}}}}\\
= \lim \dfrac{{p\left( {1 - {p^n}} \right)\left( {1 - q} \right)}}{{q\left( {1 - p} \right)\left( {1 - {q^n}} \right)}}\\
= \dfrac{{p.\left( {1 - 0} \right)\left( {1 - q} \right)}}{{q\left( {1 - p} \right).\left( {1 - 0} \right)}} = \dfrac{{p\left( {1 - q} \right)}}{{q\left( {1 - p} \right)}}
\end{array}$
