The given sequence is $a_n = \frac{n+1}{n^2+1}$, and we need to:

1. Write the first five terms of the sequence.
2. Find the limit of the sequence as $n \to \infty$, if it exists.

Step 1: First Five Terms of the Sequence

We will calculate the first five terms by substituting $n = 1, 2, 3, 4, 5$ into the formula:

• For $n = 1$: $a_1 = \frac{1+1}{1^2+1} = \frac{2}{2} = 1$

• For $n = 2$: $a_2 = \frac{2+1}{2^2+1} = \frac{3}{4+1} = \frac{3}{5}$

• For $n = 3$: $a_3 = \frac{3+1}{3^2+1} = \frac{4}{9+1} = \frac{4}{10} = 0.4$

• For $n = 4$: $a_4 = \frac{4+1}{4^2+1} = \frac{5}{16+1} = \frac{5}{17}$

• For $n = 5$: $a_5 = \frac{5+1}{5^2+1} = \frac{6}{25+1} = \frac{6}{26} = \frac{3}{13}$

Thus, the first five terms are: $a_1 = 1, \quad a_2 = \frac{3}{5}, \quad a_3 = 0.4, \quad a_4 = \frac{5}{17}, \quad a_5 = \frac{3}{13}$

Step 2: Finding the Limit of the Sequence

Now, let's find the limit of the sequence $a_n = \frac{n+1}{n^2+1}$ as $n \to \infty$.

To analyze this, consider