Question

Does the sequence {a_n} converge or diverge? Find the limit if the sequence is convergent.
a_n=[(n+4)/5n][1-(4/n)]

Answer 1

One way to do this is to simplify the expression.
a_n = (n+4)/(5n) - (4n + 16)/(5n^2)
= 1/5 + 4/(5n) - 4/(5n) - 16/(5n^2)
= 1/5 - 16/(5n^2)

So a_n = 1/5 - 16/(5n^2).

So as n goes to infinity, we see a_n approaches 1/5.