Q) Find $$\lim_{n\to \infty}\int_{(0,\infty)}\frac{dt}{\left(1+\frac{t}{n}\right)^nt^{1/n}}$$
I was hoping to apply Dominated convergence theorem to use $$\lim_{n\to \infty}\left(1+\frac{t}{n}\right)^{-n}= e^{-t}$$
but clearly I know the lower bounds but not an upper bound function which is absolutely integrable.
$$\left(1+\frac{t}{n}\right)^{-n}\geq e^{-t},t^{-1/n}\geq t^{-1} \quad\text{at least on } (1,\infty)$$
Can I find an upper bound or have to figure out the integral itself first? Thanks.
Best Answer
On $(0,1]$, the sequence $\{t^{-1/n}\}_{n=1}^{\infty}$ is decreasing. And $\{(1+t/n)^{n}\}_{n=1}^{\infty}$ is increasing for all $t>0$, and $\{(1+t/n)^{-n}\}_{n=1}^{\infty}$ is decreasing.
Note that $\displaystyle\int_{0}^{1}\dfrac{1}{(1+t/2)^{2}}\dfrac{1}{t^{1/2}}dt\leq\int_{0}^{1}\dfrac{1}{t^{1/2}}dt<\infty$, so by Monotone Convergence Theorem $\displaystyle\int_{0}^{1}\dfrac{1}{(1+t/n)^{n}}\dfrac{1}{t^{1/n}}dt\rightarrow\int_{0}^{1}e^{-t}dt$.
Note that Monotone Convergence Theorem has the decreasing version, as long as $f_{1}\in L^{1}$, $f_{1}\geq f_{2}\geq\cdots$, then the integrals correspond to them also converge.
Now we look at the interval $[1,\infty)$. We have \begin{align*} \dfrac{1}{(1+t/n)^{n}}\dfrac{1}{t^{1/n}}\leq\dfrac{1}{(1+t/n)^{n}}\leq\dfrac{1}{(1+t/2)^{2}} \end{align*} and $\displaystyle\int_{1}^{\infty}\dfrac{1}{(1+t/2)^{2}}dt<\infty$, an integrable upper bound is obtained.