How to find a limit of this sequence: $\lim\limits_{n \to \infty} \sum\limits_{k=1}^n \frac{1}{\sqrt{kn}}$

summation calculus limits radicals

How to fing a limit $$\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{\sqrt{kn}}?$$ I had only two minds about this. First of them was that it looks like $\frac{1^k +…+n^k}{n^{k+1}}$ which limit is $\frac{1}{k+1}$. The second is to look at square of $1+\frac{1}{\sqrt{2}} + .. + \frac{1}{\sqrt{n}}$, but it is a kind of a monster. I think, I need a formula for $1+\frac{1}{\sqrt{2}} + .. + \frac{1}{\sqrt{n}}$, but I don’t know it.


HINT

We have that

$$\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{\sqrt{kn}}=\lim_{n \to \infty} \frac1n\sum_{k=1}^n \frac{1}{\sqrt{\frac k n}}$$

then refer to

  • Perfect understanding of Riemann Sums

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