Question

does the sequence \\( \\{ a _ { n } \\} \\) converge or diverge? find the limit if the sequence is convergent.

\\( a _ { n } = \frac { n ^ { 2 } + 2 n + 1 } { n + 1 } , n \geq 0 \\)

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

\\( \bigcirc \\) a. the sequence converges to \\( \lim _ { n \
ightarrow \infty } a _ { n } = \\) (simplify your answer.)

\\( \bigcirc \\) b. the sequence diverges.

Explanation:

Step1: Simplify the sequence

We know that \(n^{2}+2n + 1=(n + 1)^{2}\). So, \(a_{n}=\frac{n^{2}+2n + 1}{n + 1}=\frac{(n + 1)^{2}}{n+1}\).
Since \(n\geq0\), \(n + 1
eq0\). Then \(a_{n}=n + 1\) for \(n\geq0\).

Step2: Find the limit as \(n\to\infty\)

We calculate \(\lim_{n
ightarrow\infty}a_{n}=\lim_{n
ightarrow\infty}(n + 1)\).
As \(n\to\infty\), \(n+1\to\infty\).

Answer:

B. The sequence diverges.