A_n = (1 + (3/n) )^4n
I don't know hot to find if this converges. I feel like I should break up the exponent, but I don't know how that would help.
This uses a couple of neat tricks.
Let this limit equal A. Then take the natural log of both sides and you get:
Now if you try to evaluate this limit for n -> infinity, you get an indeterminate form, so you can use L'Hopitals rule.
The right hand side stays the same because after we apply the rule, nothing changes about the limit.
Now it's just simplifying and you should get it down to:
Raise both sides to the power of e and you get