Question

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

\\( a _ { n } = \frac { 25 n ^ { 2 } + 30 n + 9 } { 5 n + 3 } , 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: Factor the numerator

$$25n^{2}+30n + 9=(5n + 3)^{2}$$
So, \(a_{n}=\frac{(5n + 3)^{2}}{5n+3}\)

Step2: Simplify the expression

Since \(n\geq0\), \(5n + 3
eq0\). Then \(a_{n}=5n + 3\) for \(n\geq0\)

Step3: Find the limit as \(n\to\infty\)

\(\lim_{n
ightarrow\infty}a_{n}=\lim_{n
ightarrow\infty}(5n + 3)\)
As \(n\to\infty\), \(5n\to\infty\) and \(5n+3\to\infty\)

Answer:

B. The sequence diverges.