Question

Determine whether the sequence:

ln(2n^2 +1) - ln(n^2 +1)

converges or diverges. If the sequence converges, find the limit.

I tried expressing the sequence as:
ln(2n^2 +1/n^2 +1), but still no luck.

The answer in the back of the book is ln(2)

I am not sure how to manipulate the equation in order to take the limit.

Can I solve this using squeeze theorum?

Answer 1

i believe this should do the trick:
\[\ln(2n^{2}+1)-\ln(n^{2}+1) = \ln\frac{2n^{2}+1}{n^{2}+1}\]
The limit of the fraction will be 2, so basically as n goes to infinity, the sequence converges to ln 2

Answer 2

if you need me to clear up anything, just ask :)

Answer 3

wouldn't the fraction end up being 2(∞)^2 +1/(∞)^2+1 be an indeterminate form in which I would need to apply l'hospitals rule?

Answer 4

ohhhh I see what you meant!

Answer 5

thanks now I get it

Answer 6

sweet :)