Question

How can i prove that this is a converged range when n->infint, and how can i find limit? an=((2^3-1)/(2^3+1))*((3^3-1)/(3^3+1))...*(n^3-1)/(n^3+1)? Thank you.

Answer 1

See proof of convergence below.

Explanation:

I interpret your question as:

`lim prod_(n=2->oo) (n^3-1)/(n^3+1)`

To show that this product converges we can first analyse the general term as `n-> oo`

`lim_(n->oo) (n^3-1)/(n^3+1) = lim_(n->oo) (1-1/n^3)/(1+1/n^3)`

`= (1-0)/(1+0) =1`

So, ultimately the product will be multiplied by 1.

Now consider for all finite `n >=2` `(n^3-1) < (n^3+1)`

This implies that each term in the product is less than 1 and hence the product of `n` terms will also be less that 1.

This result, combined with the limit of the general term above, proves that the infinite product converges to a limit less than 1.

NB: I have not yet found an analytic method to determine the actual limit. I have applied numerical methods to approximate the limit. This approximates to `0.66667` to 5 sf. Given this, I expect, but have not proved, that the limit is `2/3`.