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I am trying to prove that the

$A_n=n\int_{1}^{+\infty}\left(1-e^{-\frac{1}{x-1}}\right) e^{-x}\left(1-e^{-x}\right)^{n-1}$ $\Rightarrow$ $\lim_{n\to\infty}A_n=0$ .

In fact, by defining $f_n(x)=\color{red}{n}\left(1-e^{-\frac{1}{x-1}}\right) e^{-x}\left(1-e^{-x}\right)^{n-1}$, it is straightforward to show that $f_n$ converges pointwise to zero function as n tends to infinity. So, if I want to use Dominated Convergence Theorem, I should show that there exists an integrable function $g(x)$ such that $|f_n(x)|\leq g(x)$ for all $n$. My problem is I cannot find $g(x)$ to satisfy the constraints. Thank you for your help and comments.

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  • $\begingroup$ For large enouh $n$ All elements are bounded by $1$ or 2. So just use $g_(x)=2e^{-x}$. $\endgroup$ – Boby Apr 20 '17 at 21:52
  • $\begingroup$ @Boby Thanks. I think this function does not work. Why 2? $\endgroup$ – Mahdi Apr 20 '17 at 22:05
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Splitting the integral might be a good start.

Given $\epsilon>0$, write $$ A_n=\int_1^{\infty} = \int_{1}^{x_n} + \int_{x_n}^{\infty}= C_n+D_n.$$ The sequence $x_n$ will be determined later so that $x_n\rightarrow\infty$ as $n\rightarrow\infty$.

We have by substitution $t=1-e^{-x}$, $dt=e^{-x}dx$, $$\begin{align} C_n&\leq n\int_{1}^{x_n} e^{-x} (1-e^{-x})^{n-1} dx = n \int_{1-e^{-1}}^{1-e^{-x_n}} t^{n-1} dt \\ &\leq (1-e^{-x_n})^n \leq \exp(-\frac{n }{e^{x_n}}),\end{align} $$ and for sufficiently large $n$, we have $$\begin{align} D_n&\leq n\int_{x_n}^{\infty} (1-e^{-\frac1{x_n-1}}) e^{-x}(1-e^{-x})^{n-1} dx \\ &\leq \frac n{x_n-1}\int_{x_n}^{\infty} e^{-x}(1-e^{-x})^{n-1}dx \leq \frac n{x_n-1}\cdot \frac1n=\frac1{x_n-1}.\end{align}$$ Take $e^{x_n} = \sqrt n$. Then $$ \lim_{n\rightarrow\infty} C_n = 0, \ \ \lim_{n\rightarrow\infty} D_n = 0.$$ Therefore, $\lim_{n\rightarrow\infty} A_n = 0$.

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  • $\begingroup$ Thanks. Your idea is brilliant. Because in my research I frequently encounter with bounding, finding limit,.... , could you introduce me books, courses or exercises to learn such a techniques? My background is in Electrical Engineering and I want to increase my math knowledge. Best. $\endgroup$ – Mahdi Apr 23 '17 at 8:53
  • $\begingroup$ Hi, if you learned calculus, then I recommend you learning real analysis. There are many textbooks on real analysis such as Marsden, Rudin, Bartle, etc. For electrical engineering, I think Marsden is a good choice because it has an emphasis on Fourier analysis and multivariable calculus. $\endgroup$ – Sungjin Kim Apr 23 '17 at 17:20
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As you were discovering, your problem is not really a DCT situation. Perhaps you have seen the following result, or something like it: Let $0<b\le 1.$ Suppose $f$ is continuous on $[0,b]$ with $f(0) = 0.$ Then

$$\lim_{n\to \infty} n\int_0^b f(x)(1-x)^{n-1}\,dx = 0.$$

Now in the given problem, make the substitution $x= \ln (1/y).$ Your expression turns into

$$n\int_0^{1/e} f(y)(1-y)^n\,dy,$$

with $f$ satisfying the hypotheses above. The result follows.

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  • $\begingroup$ This is nice to see a general theorem. (+1). The idea of proof is splitting the integral which is in my answer. $\endgroup$ – Sungjin Kim Apr 21 '17 at 23:56

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