Question

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.

Answer 1

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.