does the sequence {an} converge or diverge? find the limit if the sequence is convergent. an = 2 - 6n^4 /…

does the sequence {an} converge or diverge? find the limit if the sequence is convergent. an = 2 - 6n^4 / n^4 + 3n^3 select the correct choice below and, if necessary, fill in the answer box to complete the choice. a. the sequence converges to lim an = (simplify your answer.) n→∞ b. the sequence diverges.

does the sequence {an} converge or diverge? find the limit if the sequence is convergent. an = 2 - 6n^4 / n^4 + 3n^3 select the correct choice below and, if necessary, fill in the answer box to complete the choice. a. the sequence converges to lim an = (simplify your answer.) n→∞ b. the sequence diverges.

Answer

Explanation:

Step1: Divide numerator and denominator by highest - power term

Divide both the numerator and denominator of $a_n=\frac{2 - 6n^4}{n^4+3n^3}$ by $n^4$. We get $a_n=\frac{\frac{2}{n^4}-6}{1 + \frac{3}{n}}$.

Step2: Find the limit as $n\to\infty$

We know that $\lim_{n\to\infty}\frac{1}{n^k}=0$ for $k>0$. So, $\lim_{n\to\infty}\frac{2}{n^4}=0$ and $\lim_{n\to\infty}\frac{3}{n}=0$. Then $\lim_{n\to\infty}a_n=\lim_{n\to\infty}\frac{\frac{2}{n^4}-6}{1+\frac{3}{n}}=\frac{0 - 6}{1+0}$.

Step3: Simplify the result

$\frac{0 - 6}{1+0}=-6$.

Answer:

A. The sequence converges to $\lim_{n\to\infty}a_n=-6$