An Application of Lebesgue Dominated Convergence Theorem

Question

As an application of the Lebesgue Dominated Convergence Theorem, we would like to evaluate the following limit $$ \lim_{n\to\infty}\int\limits_1^{n}\frac{1}{\displaystyle\left(1+\frac{x}{n}\right)^nx^{1/n}}\ \mathrm dx. $$ We realized that to put the limit the inside of the integral we need to find an integrable function $g(x)$ such that $$ \left|\frac{1}{\displaystyle\left(1+\frac{x}{n}\right)^nx^{1/n}}\right| \leq g(x). $$

We did this. But the boundary of the integral also depends on $n$. At this point, we do not know what we need to do.

Thanks in advance for any help.

Answer 1

Hint: Using $$ (1+z)^n\ge \binom{n}{2}z^2 $$ for $z>0$, one has $$ (1+\frac{x}{n})^n\ge\frac{n(n-1)}{2}\frac{x^2}{n^2}=\frac12(1-\frac1n)x^2\ge\frac14x^2$$ for $n\ge2$. Then define $$ f_n(x)=\frac{1}{(1+\frac{x}{n})^nx^{1/n}}\,1_{[0,n]} $$ to estimate $f_n$ as $$ |f_n(x)|\le\frac{4}{x^2} $$ for $x\ge1$ and then one can use LDC.

Answer 2

You have $$ \int_1^{n}\frac{1}{(1+\frac{x}{n})^nx^{1/n}}dx =\int_1^{\infty}\frac{1}{(1+\frac{x}{n})^nx^{1/n}}\,1_{[0,n]}\,dx. $$ If your $g$ does not depend on $n$, you are done.