how do u find the interval for this series after knowing that it's convergent?

Question

(x^n) / (7^n) with
inf
E
n=1

and how do u know if it's ( ) or [ ]

Answer 1

Use the Ratio Test:

lim as n -> ∞ | [ x^(n+1) / 7^(n+1) ]/[ x^n / 7^n ] |
= |x / 7|

So we have convergence for | x / 7 | < 1, that is, for |x| < 7.

Convergence at the endpoints of the interval generally needs to be determined on a case-by-case basis. Here it is easy: the nth term does not go to zero for either x=7 or x=-7, a necessary (though not sufficient) condition for convergence. So the series does not converge at either endpoint.