Question

Determine whether the following sequence converge or diverge. Find the limit if it converges

an= n ln(n-2/n)

So I think the limit goes to infinity. This would mean that the sequence is divergent, right?

Answer 1

\[an= n \ln (n-2/n)\]

Answer 2

Is it n-(2/n) or (n-2)/n?

Answer 3

(n-2)/n

Answer 4

So I did \[\lim_{n \rightarrow \infty }n \times \ln ( \lim_{n \rightarrow \infty}(n-2)/n)\]

Answer 5

I found that the limit on the right hand side is 1, and the left goes to infinity

Answer 6

lim((n-2)/n) = 1

ln(1) = 0

Answer 7

but isn't infinity times 0 undefined?

Answer 8

im not sure, i believe it would go to zero

Answer 9

No, 0 * infinity is indeterminate. Which basically means you have to find a way to solve the limit without falling into that "indetermination"

Answer 10

hmm, I see

Answer 11

you can use L'Hopitals rule
first write it as
\[\lim_{? \rightarrow ?} \frac{\ln(\frac{n-2}{n})}{\frac{1}{n}}\]
Then differentiate top and bottom
Then re-evaluate the limit

Answer 12

goes to 0?

Answer 13

No goes to -2 if i did my math right

Answer 14

dumbcow is right, it goes to -2.

Answer 15

is the natural log part a common limit?