For the sequences below, find if they converge or diverge. If they converge, find the limit.
an=(3^(n+2))/(5^n)
an=cos(2/n)
an=(2^(1+3n))^1/n
I am unsure about how to get started on these problems so some assistance would be great.
1)
(3^(n+2))/(5^n) =
3^2 (3/5)^n
This converges to zero.
2) cos(2/n) converges to cos(0) = 1
3)
(2^(1+3n))^1/n = 2^[1/n + 3] converges to 2^3 = 8.
I think 1) is obvious, but you still have to practice proving this rigorously. So, you have to show that for every epsilon there exists an N such that for all n>N, a_n is closer to the limiting value (in this case zero) than epsilon.
In 2) and 3) we use that if f(x) is a continuous function then you can take the limit inside the argument of f(x). So, in 2 you can take the limit of 2/n and then apply the cosine to the limit of zero.
You should prove this rule using he defintion of continuity and the definition of the limit.