I was wondering if someone could show me how to do this.
I need to find an N(epsilon) for the following sequences such that for all n>=N(epsilon) the absolute value of the nth element is less than epsilon >0:
I think I know how to do this. For e>0, 1/sqrt(n) > e. Thus, 1>sqrt(n)*e and (1/e) < sqrt(n). Square both sides and (1/e^2) < n. I believe this is an N(e) such that n >= N(e). Is this right?
2.) (1 + sqrt(n)) / (n^3)
I want to say that ((1 + sqrt(n)) / (n^3)) >= (1/n^3) > e. And then this leads to n > (1/e^(1/3)). Is this right?
3.) (sin(n)) / (2 + n^(5/3))
4.) (sqrt((n^4) +4) - n^2) * n
I'm not sure what to do on these. Any suggestions would be helpful.