How to integrate $xe^x$ without using antiderivatives or integration by parts.

Question

Yesterday, I sat for my Real Analysis II paper. There I found a question asking to integrate $\displaystyle\int_0^1 xe^x \, dx$ without using antiderivatives and integrating by parts.

I tried it by choosing a partition

$$P_n=(0,\frac{1}{n},\frac{2}{n},\ldots,\frac{n-1}{n},1),$$

but I was not able to show that $\displaystyle \lim_{n \to \infty} U(f,P_n)=\lim_{n \to \infty} L(f,P_n)=1$

Answer 1

We have

$$U(f,P_n) = \frac1{n^2} \sum_{k=1}^n k e^{k/n}= L(f,P_n) + \frac{e}{n}.$$

If the limit of the upper sum exists, then it is identical to the limit of the lower sum.

Note that

$$\sum_{k=1}^nk r^k = \frac{r-r^{n+1}}{(1-r)^2}- \frac{nr^{n+1}}{1-r}.$$

Using $r = e^{1/n}$ we have as $n \to \infty$

$$U(f,P_n)= \frac{1/n}{1-e^{1/n}}\frac{1/n}{1-e^{1/n}}e^{1/n}(1-e)- \frac{1/n}{1-e^{1/n}}e^{1/n}e \to 1,$$

since

$$\lim_{n \to \infty} \frac{1/n}{1-e^{1/n}}= -1$$

Answer 2

My attempt was wrong. Fixing it with my choice of partition unnecessarily complicates the calculation, especially when a solution exists with the partition the OP selected.

SUGGESTION: What would happen if you change the partition to $P_n=[0,\frac{1}{n},\frac{1}{n-1},\ldots,\frac{1}{3},\frac{1}{2},1]$?

Answer 3

My answer:

$U(f,P_n)=\frac{1}{n^2}\left(e^{\frac{1}{n}}+2e^{\frac{2}{n}}+3e^{\frac{3}{n}}+...+ne^1\right)=\frac{1}{n^2}\displaystyle\sum_{k=1}^{n}ke^{\frac{k}{n}}=\frac{1}{n^2}\displaystyle\sum_{k=1}^{n}k\left(1+\frac{k}{n}+\frac{k^2}{2!n^2}+\frac{k^3}{3!n^3}+..\right)=\frac{1}{n^2} \displaystyle\sum_{k=1}^{n}k+\frac{1}{n^2}\displaystyle\sum_{k=1}^{n}k\left(\frac{k}{n}+\frac{k^2}{2!n^2}+\frac{k^3}{3!n^3}+..\right)=\frac{n(n+1)}{2n^2}+\frac{n(2n+1)(n+1)}{2!6n^3}+\frac{n^2(n+1)^2}{3!4n^4}+....$

Thus when $n\to \infty $ , $U(f,P_n)=\frac{1}{2}+\frac{1}{2!6}+\frac{1}{3!4}+\frac{1}{4!30}+..$ should be 1