Simplifiying an equation

Question

The equation I have is this: My question is, since $\lim_{n \to \infty} {1 \over n}$ = 0, won't this whole expression be equal to 0?

$\lim_{n\to \infty}$ ${1 \over n}\sum_{k=1}^n \arcsin({k \over n})$

Answer 1

It is true that the first factor tends to zero, but you must also consider that the second factor may grow fast enough to "outdo" that.

If the second factor is bounded as $n\to\infty$, your conclusion is correct. Otherwise, you must try to analyze the growth rate of the second factor.