Determine whether the following sequences converge or diverge. Justify your answers (show why)

Question

1. a(n) = (e^n)/(3^n)
2. a(n) = (-1)^(2n+1)
3.a(n) = (ln[(e^4)^n])/3n
4. a(n) = (e^n+e^-n)/(e^(2n)-1)

Answer 1

#1 (e/3)^n clearly converges, since e/3 < 1
#2 2n+1 is always odd, so ...
#3 ln[(e^4)^n] = ln[e^(4n)] = 4n ... 4n/3n > 1
#4 e^n + e^-n = (e^2n + 1)/e^n
(e^(2n)+1)/(e^n(e^(2n)-1)) < 1 for n>1