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

Jun 14, 2015

That sum diverges.

Explanation:

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

Since:

${\lim}_{n \rightarrow + \infty} {a}_{n} = {\lim}_{n \rightarrow + \infty} {5}^{n} / \left({3}^{n} + {4}^{n}\right) =$

${\lim}_{n \rightarrow + \infty} {5}^{n} / {4}^{n} = {\lim}_{n \rightarrow + \infty} {\left(\frac{5}{4}\right)}^{n} = + \infty$

(${3}^{n}$ is negligible respect ${4}^{n}$)

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