# Thread: Proving that a_n = (n^3-1)/(n^3+1) converges to 1

1. ## Proving that a_n = (n^3-1)/(n^3+1) converges to 1

Hi, I have a very simple example of 1/n converging to 0 (it cannot be solved using a limit, that would be very easy, we have to show it using the definition of the limit), but I don't know how to prove a more complex sequence using the definition of the limit.

The problem says:

"Using only the definition of the limit prove that the sequence a_n = (n^3-1)/(n^3+1) converges to 1."

I would really appreciate it if you could show me how to solve this, thanks a lot!

2. ## Re: Proving that a_n = (n^3-1)/(n^3+1) converges to 1

Given $\displaystyle \epsilon$, you need to find for which n we have $\displaystyle 1 - a_n < \epsilon$ ($\displaystyle a_n < 1$, so this implies $\displaystyle |1 - a_n| < \epsilon$ required by the definition of limit). Just solve this inequality.

3. ## Re: Proving that a_n = (n^3-1)/(n^3+1) converges to 1

note that $\displaystyle \frac{n^3-1}{n^3+1} = 1-\frac{2}{n^3+1}$ so

$\displaystyle |a_n-1| = \frac{2}{n^3+1}$

if we want this to be less than epsilon, how shall we choose n?