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I wish to find: $$\lim_{n\to\infty} n\int_0^1 (1+x)^{-n-1}e^{x^2}\ dx\ \ ( > n=1,2,\cdots)$$

I maybe have to evaluate: $$\int_0^1 (1+x)^{-n-1}e^{x^2}\ dx$$

But I can't, can someone give me some help?

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3 Answers 3

up vote 3 down vote accepted

We can partially evaluate the integral, by integrating by parts,

$$\begin{align} \int_0^1 \frac{n}{(1+x)^{n+1}}e^{x^2}\,dx &= \left[-\frac{e^{x^2}}{(1+x)^n} \right]_0^1 + \int_0^1 \frac{2xe^{x^2}}{(1+x)^n}\,dx\\ &= 1 - \frac{e}{2^n} + \int_0^1 \frac{2xe^{x^2}}{(1+x)^n}\,dx \end{align}$$

The last two terms converge to $0$, so the limit is $1$.

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Let $u=nx$ then we have $$n\int_0^1 (1+x)^{-(n+1)}e^{x^2}dx=\int_0^n(1+u/n)^{-(1+n)}e^{(u/n)^2}du\to\int_0^\infty e^{-u}du=1$$

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$\newcommand{\angles}[1]{\left\langle #1 \right\rangle}% \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}% \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}% \newcommand{\ds}[1]{\displaystyle{#1}}% \newcommand{\expo}[1]{{\rm e}^{#1}}% \newcommand{\ic}{{\rm i}}% \newcommand{\imp}{\Longrightarrow}% \newcommand{\pars}[1]{\left( #1 \right)}% \newcommand{\pp}{{\cal P}}% \newcommand{\ul}[1]{\underline{#1}}% \newcommand{\verts}[1]{\left\vert #1 \right\vert}$ The main contribution comes from values of $x \gtrsim 0$. Then, \begin{align}\color{#ff0000}{\large% \lim_{n \to \infty}n\int_{0}^{1}\pars{1 + x}^{-n - 1}\expo{x^{2}}\,{\rm d}x} &= \lim_{n \to \infty}n\int_{0}^{1}\expo{-\pars{n + 1}\ln\pars{1 + x}}\,{\rm d}x \\[3mm]&= \lim_{n \to \infty}n\int_{0}^{1}\expo{-\pars{n + 1}x}\,{\rm d}x = \lim_{n \to \infty}{n \over n + 1}\int_{0}^{n + 1}\expo{-x}\,{\rm d}x \\[3mm]&= \int_{0}^{\infty}\expo{-x}\,{\rm d}x = \color{#ff0000}{\Large 1} \end{align}

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