I need help with this exercise on limits at infinity.

Solution 1:

You have $a_p=-a_1-a_2-\cdots-a_{p-1}$ and therefore your limit is the sum of the limits\begin{multline}\lim_{n\to\infty}a_0\sqrt{n}-a_0\sqrt{n+p},\lim_{n\to\infty}a_1\sqrt{n+1}-a_1\sqrt{n+p},\ldots\\\ldots,\lim_{n\to\infty}a_{p-1}\sqrt{n+p-1}-a_{p-1}\sqrt{n+p}\end{multline}(if all of them exist). And it's not hard to prove that each of them is equal to $0$.

Solution 2:

Hint I think that your idea is almost right, just that you shouldn't consider $\sqrt{n}$ as a factor you have to pull out, but as a summand:
Observe that for a given $n$, we have

$$ a_0 \sqrt{n} + a_1 \sqrt{n+1} + \dots + a_p \sqrt{n+p} = \sum_{i=0}^{p} a_i \sqrt{n} + \sum_{i=0}^{p} a_i (\sqrt{n+i} - \sqrt{n}) $$

Can you show that this converges to $0$?