00:01
We want to determine whether the sequence a -n is equal to hyperbolic tangent of n converges or diverges, and if it converges what it converges.
00:12
Now, before we actually try to take the limits of anything, let's recall that hyperbolic tangent is rarely e to the end minus e to the negative n over e to the n.
00:34
Minus or plus e to the negative end.
00:38
Now if we were to go ahead and just apply in goes to infinity, well we get infinity minus zero over infinity minus zero, which would be infinity over infinity.
01:01
And while this is the setup for using loophytols, before we do that, we can try to simplify this a little bit by factoring out a e to the minus n out of the top and bottom, and doing that we would be left with e to the 2n minus 1 over e to the 2n plus 1.
01:31
And so this here would still give us, so as n goes to infinity, it would still be infinity minus 1 over infinity plus 1, which still goes to infinity over infinity.
01:45
So we can still use lopatol's, but taking the derivative of this will be a little bit more straightforward.
01:53
So now we have the limit as an approach to infinity of a .n is going to equal to, and this is going to be by lopatol's...