Question

How do you determine if the sum of `5^n/(3^n + 4^n)` from n=0 to infinity converges?

Answer 1

That sum diverges.

Explanation:

Since it is a sum of all positive numbers, it is regular (convergent or divergent, not irregular).

Since:

`lim_(nrarr+oo)a_n=lim_(nrarr+oo)5^n/(3^n+4^n)=`

`lim_(nrarr+oo)5^n/4^n=lim_(nrarr+oo)(5/4)^n=+oo`

(`3^n` is negligible respect `4^n`)

and it is not `0`, then it is divegent.