Question

2.an=1/2n 3.añ=1+2n.4.an=1+(_1)ñ/n 5. an=sin n6. an=2n7.an=[n+(-1)ñ]‐¹ 8.(n÷n-1)²​

Answer 1

Answer:

{x6n} is a subsequence of {x2n} and {x3n} so it converges to the same limit as both of them. Similarly, {x6n+3} is a subsequence of the odds and {x3n} so it converges to the same limit as both of them. So all three sequences converge to the same limit.

For every ϵ>0 we can find N1 and N2 such that if 2n>N1 then |x2n−l|<ϵ and if 2n+1>N2 then |x2n−l|<ϵ