Giải thích các bước giải:
\(\begin{array}{l}
43,\\
S = 1.2 + 2.3 + 3.4 + .... + n\left( {n + 1} \right)\\
\Leftrightarrow 3S = 1.2.3 + 2.3.3 + 3.4.3 + ..... + n.\left( {n + 1} \right).3\\
\Leftrightarrow 3S = 1.2.\left( {3 - 0} \right) + 2.3.\left( {4 - 1} \right) + 3.4.\left( {5 - 2} \right) + ..... + n.\left( {n + 1} \right).\left[ {\left( {n + 2} \right) - \left( {n - 1} \right)} \right]\\
\Leftrightarrow 3S = 1.2.3 - 0.1.2 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + ..... + n.\left( {n + 1} \right)\left( {n + 2} \right) - \left( {n - 1} \right).n.\left( {n + 1} \right)\\
\Leftrightarrow 3S = n\left( {n + 1} \right)\left( {n + 2} \right) - 0.1.2\\
\Leftrightarrow S = \frac{{n\left( {n + 1} \right)\left( {n + 2} \right)}}{3}\\
\lim \frac{{1.2 + 2.3 + 3.4 + ..... + n\left( {n + 1} \right)}}{{ - {n^3} + {n^2} - 1}}\\
= \lim \frac{{n\left( {n + 1} \right)\left( {n + 2} \right)}}{{3.\left( { - {n^3} + {n^2} - 1} \right)}}\\
= \lim \frac{{{n^3} + 3{n^2} + 2n}}{{ - 3{n^3} + 3{n^2} - 3n}}\\
= \lim \frac{{1 + \frac{3}{n} + \frac{2}{{{n^2}}}}}{{ - 3 + \frac{3}{n} - \frac{3}{{{n^2}}}}}\\
= \frac{{ - 1}}{3}\\
46,\\
{1^3} + {2^3} + {3^3} + ..... + {n^3} = {\left[ {\frac{{n\left( {n + 1} \right)}}{2}} \right]^2}\\
\lim \frac{{{1^3} + {2^3} + .... + {n^3}}}{{ - {n^5} + 4n - 4}}\\
= \lim \frac{{{{\left[ {n\left( {n + 1} \right)} \right]}^2}}}{{4.\left( { - {n^5} + 4n - 4} \right)}}\\
= \lim \frac{{{n^2}\left( {{n^2} + 2n + 1} \right)}}{{ - 4{n^5} + 16n - 16}}\\
= \lim \frac{{{n^4} + 2{n^3} + {n^2}}}{{ - 4{n^5} + 16n - 16}}\\
= \lim \frac{{\frac{1}{n} + \frac{2}{{{n^2}}} + \frac{1}{{{n^3}}}}}{{ - 4 + \frac{{16}}{{{n^4}}} - \frac{{16}}{{{n^5}}}}}\\
= \frac{0}{{ - 4}} = 0
\end{array}\)
