Evaluating a limit by Riemann sums I'm having some issues with understanding exactly how to see that a given limit/sum is a Riemann sum. For example: 
$\lim_{n \to \infty} \frac{1}{\sqrt{n}} \sum_{k=1}^n \frac{1}{\sqrt{k}}$.
I recognize that $\frac{1}{\sqrt{n}}$ is the length of the partitions, and the other fraction is $f(c_k)$, where $c_k$ is a value picked in the k'th interval of the n'th partition, but to me it looks like $f(x) = \frac{1}{\sqrt{x}}$ and so $c_k$ should be just $k$, but $k$ is a lot bigger than most of the partitions' length. 
Another type of problem I see is where $c_k$ depends on $n$, and again it seems to me that there isn't a $c_k$ for each k'th interval. For example this problem:
$\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n sin(\frac{k\pi}{2n})$.
Here the length of each partition is $\frac{1}{n}$, but $c_k = \frac{k\pi}{2} \frac{1}{n}$, and since the "multiplier" is greater than one, it looks to me like $c_k$ is bound to skip some intervals eventually, and so there isn't a $c_k$ to each interval, and then it's no longer a Riemann sum. 
But, obviously it is, I just don't understand it.. can anyone explain this to me? My textbook has a few problems like this, but no explanations. Thanks a lot in advance.

Look at ${1 \over n} \sum_{k=1}^n \sqrt{n \over k} = {1 \over n} \sum_{k=1}^n {1 \over \sqrt{k \over n}} ={1 \over n} \sum_{k=1}^n f({k \over n})  $, where
$f(x) = {1 \over \sqrt{x}}$.
