Question

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

\\( a _ { n } = \frac { 1 + 4 n ^ { 4 } } { n ^ { 4 } + 2 n ^ { 3 } } \\)

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

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

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

Explanation:

Step1: Divide numerator and denominator by \(n^{4}\)

$$\lim_{n ightarrow\infty}a_{n}=\lim_{n ightarrow\infty}\frac{\frac{1}{n^{4}} + 6}{\ 1+\frac{2}{n}}$$

Step2: Apply the limit

As \(n
ightarrow\infty\), \(\lim_{n
ightarrow\infty}\frac{1}{n^{4}} = 0\) and \(\lim_{n
ightarrow\infty}\frac{2}{n}=0\)

$$\lim_{n ightarrow\infty}\frac{\frac{1}{n^{4}} + 6}{\ 1+\frac{2}{n}}=\frac{0 + 6}{1+0}$$

Answer:

A. The sequence converges to \(\lim_{n
ightarrow\infty}a_{n}=6\)