Consider the convergent sequence with terms defined by

Let . Find the value of and values of such that for all for each of the following values of :

- ,
- ,
- ,
- ,
- .

First, we know

So then we have,

Thus, if then for every we have . We compute for the given values of as follows:

- implies .
- implies .
- implies .
- implies .
- implies .

2n/(n^3+1) 2/epsilon

2n/(n^3+1) < 2n/n^3 = 2/n^2 = 2/epsilon

This blog’s comment system is so bugged. I will try to use words instead, remove the one from the denominator, then cancel the ns and get reduse the power from 2 to 1, then n greater than two over epsilon works.

Dear sir/madam,

please clarify me the last step.Here N refers to a natural number.But we havent arrived at the appropriate values of N

Any integer $N$ greater than the given values will work.

I still don’t see why we can accept the last step :(