For this question, I can't really seem to get the final answer. I get stuck on the step right before you take the limits. Can anyone please help me out?
$$\begin{align}\lim_{n\to \infty}\sum_{k=0}^n (k^2+k+1) & =\lim_{n\to\infty}(\sum_{k=0}^n k^2 + \sum_{k=0}^n k + \sum_{k=0}^n 1)\\ & =\lim_{n\to \infty}(\frac{n(2n+1)(n+1)}{6} + \frac{n(n+1)}{2} + n)\\ & = \lim_{n\to \infty}(\frac{2n^3+6n^2+10n}{6}) \end{align}$$
On
For the final calculation of the final limit expression, it simply is :
$$\lim_{n\to \infty}\bigg(\frac{2n^3+6n^2+10n}{6}\bigg)=\lim_{x\to \infty} \frac{1}{3}n^3 = \infty$$
There is a rule for polynomial limits to infinity and it states that the limit of a polynomial at $\infty$ is simply the limit of the highest power term. This happens because the bigger the power gets, the biggest the impact it has on a really big value.
Simply note that
$$\frac{2n^3+6n^2+10n}{6} \ge\frac{0+0+10n}{6} = \frac53n\ge n\to +\infty$$