Question

Solve ${lim}_{x\to \infty }\left({\sum }_{k=1}^{1000}\frac{{(x+k)}^m}{x^m+{10}^{1000}}\right)$

Answer 1

${lim}_{x\to \infty }\left({\sum }_{k=1}^{1000}\frac{{(x+k)}^m}{x^m+{10}^{1000}}\right)={\sum }_{k=1}^{1000}\left({lim}_{x\to \infty }\frac{{(x+k)}^m}{x^m+{10}^{1000}}\right)$

$={\sum }_{k=1}^{1000}\left({lim}_{x\to \infty }\frac{{(1+\frac{k}{x})}^m\cdot x^m}{(1+\frac{{10}^{1000}}{x^m})\cdot x^m}\right)$

$={\sum }_{k=1}^{1000}\left({lim}_{1/x\to 0}\left(\frac{{(1+\frac{k}{x})}^m}{(1+\frac{{10}^{1000}}{x^m})}\right)\right)$

$={\sum }_{k=1}^{1000}\left(\frac{{(1+0)}^m}{(1+0)}\right)$

$=1000$