What is the Limit of the Sequence {[(n+3)/(n+1)]^n} as n Approaches Infinity?

[Beamsbox]

Basically,

find the limit of the sequence:

{[(n+3)/(n+1)]^n}, from n=1 to infinity

Book says it's supposed to be e^2, and indeed the graph shows that... I'm not sure what to do with the top of the fraction. Working with the bottom and dividing by n, I obtain, lim as n approaches infinity, (1+(1/n))^n, which is the definition of e... but I'm not sure of the legality of dividing the top and bottom by n, as they're inside the parenthesis to begin with... but if I do it to the top too, I get lim (1+3/n)^n, which I'm not sure what to do with...
lost...

Any help much appreciated!

 

[LCKurtz]

Your title is wrong; this isn't an infinite series, it is a sequence. And although it isn't written that way, I assume it is the whole fraction that is raised to the nth power, not just the denominator.

Hint: Do long division on the fraction on the inside to write it as 1 + (..) and try to get it in the form

(1 + 1/t)^t.

 

[Beamsbox]

Right, nice assumption, edited and fixed. Long division, I knew I needed it in the form ofthe definition of e, but didn't know how... I'll check it out. Thanks for the help.