points) Determine if the following sequence converges. If so, find the limit. $$ a_{n}=\frac{3 n^{2}+2 n+1}{n^{2}+1} $$

Question

UpStudy AI · Accepted Answer

We are given the sequence

$$
a_n = \frac{3n^2 + 2n + 1}{n^2 + 1}.
$$

1. **Divide numerator and denominator by $$ n^2 $$:**

$$
   a_n = \frac{3 + \frac{2}{n} + \frac{1}{n^2}}{1 + \frac{1}{n^2}}.
   $$

2. **Take the limit as $$ n \to \infty $$:**

As $$ n \to \infty $$, the terms $$ \frac{2}{n} $$ and $$ \frac{1}{n^2} $$ approach 0. Therefore, we have

$$
   \lim_{n \to \infty} a_n = \frac{3 + 0 + 0}{1 + 0} = 3.
   $$

Thus, the sequence converges, and its limit is

$$
\boxed{3}.
$$
The sequence converges to 3.

points) Determine if the following sequence converges. If so, find the limit.