Sunday, 26 February 2017

calculus - Find limlimitsnrightarrowinftyint10left(1+fracxnright)ndx


Find the limit of



limn10(1+xn)ndx




Let u= 1 +\frac{x}{n} \implies du =\frac{1}{n} dx \implies n \cdot du = dx
at x=0 u=1 and at x=1 u=1+\frac{1}{n} so now limit will change from 1 to 1+\frac{1}{n}



Back to the integral




\lim\limits_{n \rightarrow \infty} \left( n \cdot \int_{1}^{1+\frac{1}{n}} u^n du \right)= \lim\limits_{n \rightarrow \infty} \left( n \cdot \left[ \frac{nu^{n+1}}{n+1} \right]_1^{1+\frac{1}{n}} \right) = \lim\limits_{n \rightarrow \infty} \left(\frac{n^2}{n+1} \left[ u^{n+1} \right]_1^{1+\frac{1}{n}} \right)



\implies\lim\limits_{n \rightarrow \infty} \left(\frac{n^2}{n+1} \left[ \left(1+\frac{1}{n} \right)^{n+1}-1 \right] \right)=\infty



Is my finding correct? Is the procedure of taking the limit before completing the integration correct?



Much appreciated

No comments:

Post a Comment

real analysis - How to find lim_{hrightarrow 0}frac{sin(ha)}{h}

How to find \lim_{h\rightarrow 0}\frac{\sin(ha)}{h} without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...