Question

Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.) an = 7n 1 + 8n lim n→[infinity] an = Incorrect:

Answer 1

The sequence diverges

Explanation:

Given the sequence:

`a_n=7n(1+8n)=7n+56n^2`

Let's use the zero test which states:

`\Sigma a_n\\Diverges\hspace{3}if\hspace{3} \lim_(n \to \infty) a_n \\eq0`

So, let's find the limit:

`\lim_(n \to \infty) 7n+56n^2= \lim_(n \to \infty) 7n + \lim_(n \to \infty) 56n^2`

For the first limit:

`\lim_(n \to \infty) 7n= 7 \lim_(n \to \infty) n=7* \infty=\infty`

For the second limit:

`\lim_(n \to \infty) 56n^2=56 \lim_(n \to \infty) n^2 =56(\infty)^2=\infty`

So:

`\lim_(n \to \infty) 7n+56n^2=\infty +\infty=\infty`

Therefore, the sequence diverges