The problem requires us to find the first five terms of the given sequences $\{a_n\}$, determine if each sequence converges or diverges, and if it converges, find its limit as $n$ approaches infinity. We will solve problem number 2: $a_n = \frac{3n+2}{n+1}$.
2025/5/27
1. Problem Description
The problem requires us to find the first five terms of the given sequences , determine if each sequence converges or diverges, and if it converges, find its limit as approaches infinity. We will solve problem number 2: .
2. Solution Steps
First, we find the first five terms of the sequence by substituting into the formula for :
Next, we find the limit of the sequence as approaches infinity:
To find this limit, we can divide both the numerator and the denominator by :
As approaches infinity, and both approach
0. Therefore,
Since the limit exists and is equal to 3, the sequence converges to
3.
3. Final Answer
The first five terms of the sequence are .
The sequence converges, and .