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.
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 $n\to\infty$.