I'm supposed to convert this limit to a definite integral but I'm not sure how I'm supposed to do it.

$$\lim_{n\to \infty}\sum_{i=1}^n \left(\frac{1}{n}\right)\cdot\sqrt{ \frac i{n}} $$

Would appreciate some help/guidelines on how to tackle problems like this because I haven't found that many great sources to learn from when Googling it. My textbook is also really vague on this and doesn't really explain it or include examples.

  • $\begingroup$ What is $\Delta x$? Once you recognize that, what is $f(x_i^*)$? From this, you can determine what $f(x)$ is and the bounds for the integral. $\endgroup$ – Clayton Feb 14 '18 at 21:02
  • $\begingroup$ It's the limit of Riemann sums for $\int_0^1f.$ What is $f$ in this case? $\endgroup$ – zhw. Feb 14 '18 at 21:05
  • $\begingroup$ I would assume that $f(x) = \sqrt x $ here? And $ \Delta x $ is equal to $ \frac 1 {n} $ ? $\endgroup$ – gbgult Feb 14 '18 at 21:08
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    $\begingroup$ For any $f\in C^0([0,1])$ we have that $$ \lim_{n\to +\infty}\frac{1}{n}\sum_{k=1}^{n}f\left(\frac{k}{n}\right) = \int_{0}^{1}f(x)\,dx. $$ In your case $f(x)=\sqrt{x}$ gives that the limit equals $\frac{2}{3}$. $\endgroup$ – Jack D'Aurizio Feb 14 '18 at 21:12
  • $\begingroup$ @JackD'Aurizio is the $f$ in the sum necessarily the same $f$ of the integral? To me it seems that only on $\mathbb{Q}$ they have to give the same picture. Since for example for no $k,n\in \mathbb{N}\quad f(k/n)=f(1/\sqrt{2}).$ I accept that the result is the same but the functions don't have to be the same, right? $\endgroup$ – ty. Feb 14 '18 at 21:43

By inspection, this looks like a Riemann sum of some function where the partition consists of $n$ evenly spaced points between $0$ and $1$. That is, if $f(x)=\sqrt{x}$, and $P=\{0,1/n,2/n,\ldots, n/n\}$, then a (uppeR) Riemann sum of this function over $P$ is $$\sum_{i=1}^n\left(\frac{i+1}{n}-\frac{i}{n}\right)f\left(\frac{i}{n}\right)=\sum_{I=1}^n\frac{1}{n}\sqrt{\frac{i}{n}}.$$

Since $f$ is integrable on $0,1$, this converges to $\int_0^1 f$ as $n\to\infty$.

  • $\begingroup$ Is there some "general things" that I should start looking for in problems like this to see what $ \Delta x $ etc is? This was a pretty easy question so I guess it won't be as simple for me going forward to more advanced stuff. Also, I don't really get how you get that $( \frac {i+1} {n} - \frac i {n}) f( \frac i {n}) $ is equal to the sigma notation in my post? $\endgroup$ – gbgult Feb 14 '18 at 22:19

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