Evaluate $\lim _{n\to \infty }\int _{0}^{1}nx^ne^{x^2}dx$

Evaluate $$\lim _{n\to \infty }\int _{0}^{1}nx^ne^{x^2}dx.$$

I applied the mean value thorem of integral to $$\int _{0}^{1}nx^ne^{x^2}dx.$$ We get $$c\in (0,1):$$

$$\int _{0}^{1}nx^ne^{x^2}dx=(1-0)nc^ne^{c^2}.$$ Taking limit ($$\lim_{n\to \infty}$$)on the both side,

We get, $$\lim_{n\to \infty}\int _{0}^{1}nx^ne^{x^2}dx=\lim_{n\to \infty} nc^ne^{c^2}=0.$$

My answer in the examination was wrong. I don't know the correct answer. Where is my mistake?

• Your $c \in (0, 1)$ depends on $n$. – sera Aug 12 at 6:37
• @sera but $f(x)=nx^ne^{x^2}$ is a continuous function on $[0,1]$ – Unknown x Aug 12 at 6:42
• @Unknownx The point $c$ is different for all $n$. You are applying the mean value theorem for integrals for different continuous functions. – Keen-ameteur Aug 12 at 6:45
• By the way, are you also asking what the correct approach to showing this would be? – Keen-ameteur Aug 12 at 6:46
• Does this answer your question? Find $\displaystyle\lim_{n\to \infty} \int_0^1 nx^n e^{x^2} dx$ – sera Aug 12 at 6:57

As the Taylor series of $$e^{x^2}$$ converges uniformly in $$[0,1]$$, $$\int_0^1n x^n\sum_{k=0}^\infty\frac{x^{2k}}{k!}dx=\int_0^1 n\sum_{k=0}^\infty \frac{x^{2k+n}}{k!}dx=\sum_{k=0}^\infty\frac n{(2k+n+1)k!}\\=\sum_{k=0}^\infty\frac 1{k!}-\sum_{k=0}^\infty\frac{2k+1}{(2k+n+1)k!}.$$

The second term vanishes because $$\sum_{k=0}^\infty\frac{2k+1}{(2k+n+1)k!}<\frac 1n\sum_{k=0}^\infty\frac{2k+1}{k!}.$$

Hence $$e.$$

• Just out of curiosity, how did you come up eith the last equality? – tommy1996q Aug 12 at 8:01
• @tommy1996q: which one ? Notice the minor typo, just fixed. – Yves Daoust Aug 12 at 8:03
• Where you equate $\frac{n}{(2k+n+1)k!}$ with those other 2 addendums, the last = you’ve written – tommy1996q Aug 12 at 14:40
• @tommy1996q: reduce to common denominator. – Yves Daoust Aug 12 at 14:41
• Oh, I’d misread it and thought it was some obscure identity. Not gonna lie, feel really dumb right now – tommy1996q Aug 12 at 20:32

Your proof is incorrect. The issue is that the $$c$$ which you choose may depend on $$n$$.

It turns out that the correct answer is in fact $$e$$. This is because for any continuous $$f : [0, 1] \to \mathbb{R}$$, we have

$$\lim\limits_{n \to \infty} \int\limits_0^1 n x^n f(x) dx = f(1)$$

This follows from the Stone-Weierstrass theorem as follows:

First, we prove that $$\lim\limits_{n \to \infty} \int\limits_0^1 n x^n f(x) dx = f(1)$$ for every $$f$$ of the form $$f(x) = x^m$$. We then extend this to all $$f$$ polynomial quite easily.

Suppose now that we have continuous $$g : [0, 1] \to \mathbb{R}$$. Given arbitrary $$\epsilon > 0$$, let $$w = \frac{\epsilon}{3}$$. Take polynomial $$f$$ s.t. $$|f - g| < w$$ (uniform norm) which is possible by Stone-Weierstrass, and take $$N$$ s.t. for all $$n \geq N$$, $$\left|\int\limits_0^1 n x^n f(x) dx - f(1)\right| < w$$. Then we have

$$$$\begin{split} \left| \int\limits_0^1 n x^n g(x) dx - g(1) \right| &\leq \left|\int\limits_0^1 n x^n g(x) dx - \int\limits_0^1 n x^n f(x) dx\right| + \left|\int\limits_0^1 n x^n f(x) dx - f(1)\right| + \left|g(1) - f(1)\right| \\[10pt] &= \left|\int\limits_0^1 n x^n (g(x) - f(x)) dx \right| + |g(1) - f(1)| + \left|\int\limits_0^1 n x^n f(x) dx - f(1)\right| \\[10pt] &< \left|\int\limits_0^1 n x^n (g(x) - f(x)) dx \right| + w + w \\[6pt] &\leq \int\limits_0^1 n x^n \left|g(x) - f(x)\right| dx + 2w \\ &\leq w \int\limits_0^1 n x^n dx + 2w \\ &= w \frac{n}{n + 1} + 2w \\[6pt] &<3w \\[6pt] &= \epsilon \end{split}$$$$

And therefore $$\lim\limits_{n \to \infty} \int\limits_0^1 n x^n g(x) dx = g(1)$$

Alternatively we could compute this limit directly via the substitution $$u = x^{n+1}$$

$$\lim_{n\to\infty} \frac{n}{n+1}\int_0^1e^{u^{\frac{2}{n+1}}}\:du \to \int_0^1e\:du = e$$

by dominated convergence.


See Laplace's Method.

A bit late answer but maybe worth noting it.

Partial integration gives

$$I_n :=\int_0^1 \underbrace{nx^{n-1}}_{u'}\cdot\underbrace{xe^{x^2}}_{v}dx= \left.x^{n+1}e^{x^2}\right|_0^1- \underbrace{\int_0^1 x^n(1+2x^2)e^{x^2}dx}_{J_n=}=e-J_n$$

Now, $$J_n$$ can be easily estimated as follows $$0\leq J_n \leq 3e\int_0^1x^ndx=\frac{3e}{n+1}\stackrel{n\to \infty}{\longrightarrow}0$$

Hence, $$I_n \stackrel{n\to \infty}{\longrightarrow} e$$.