# Find $\lim_{n \rightarrow \infty} \int_0 ^{\infty} \left(1+ \frac{x}{n}\right)^n e^{-ex} dx$ [duplicate]

Find $$\lim_{n \rightarrow \infty} \int_0 ^{\infty} \left(1+ \frac{x}{n}\right)^n e^{-ex} dx.$$

I know that $$\lim_{n \rightarrow \infty} \left(1+ \frac{x}{n}\right)^n = e^x.$$

Also, I showed that $$\left(1+ \frac{x}{n}\right)^n \leq e^x$$ for all $$x \geq -n$$.

So I'm thinking about the Lebesgue Dominated Convergence Theorem. But I'm not sure how to deal with the $$e^{-ex}$$ part.

• How about $g(x) := e^{x }e^{-ex} = e^{(1 - e)x}$ being the dominating function? Since $1 - e < 0$, the integral of $g$ does exist. Commented Feb 8, 2021 at 3:54

Let $$f_n : [0, \infty) \to \Bbb R$$ be defined by $$f_n(x) := \left(1 + \frac{x}{n}\right)^ne^{-ex}.$$
With a clever use of AM-GM, one can show that $$f_n(x) \le f_{n+1}(x)$$ for all $$x \ge 0$$. (You only need to show the inequality for the part without the $$e^{-ex}$$.)
$$\int \left(1+ \frac{x}{n}\right)^n e^{-ex}\, dx= {n^{-n}}\int(x+n)^ne^{-ex}\, dx$$ $$\int (x+a)^b e^{-c x}\, dx=-e^{a c} c^{-b-1} \Gamma (b+1,c (x+a))$$ $$\int_0^\infty (x+a)^b e^{-c x}\, dx=e^{a c} c^{-b-1} \Gamma (b+1,a c)\quad \text{if}\quad \Re(c)>0\land \Re(a)>0$$ All of that makes $$I_n=\int_0^\infty \left(1+ \frac{x}{n}\right)^n e^{-ex}\, dx=e^{(e-1) n-1}\, n^{-n} \,\Gamma (n+1,e n)$$
Have a look here [at $$§8.11(iii)$$] for $$\lambda=e$$.
Let $$f_n(x)=\left(1 + \frac{x}{n}\right)^ne^{-ex} \chi_{[0,n]}(x).$$ Then $$|f_n(x)|\le e^{-(e-1)x}, \lim_{n\to\infty}f_n(x)=f(x)=e^{-(e-1)x}.$$ Since $$\int_0^\infty e^{-(e-1)x}dx<\infty$$ the DCT, one has $$\lim_{n\to\infty}\int_0^\infty f_n(x)dx=\int_0^\infty\lim_{n\to\infty} f_n(x)dx.$$