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:
1.) 1/sqrt(n)
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.
With all of these try finding a strictly decreasing sequence that bounds the
given sequence, then apply the method used for part one to the bounding
sequence the you find for the bounding sequence will then suffice
for the given sequence.
By the way (4) is divergent so you will not be able to find such an .
RonL