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

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Answer 1 · 2015-06-14T18:44:19

That sum diverges.

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.

How do you determine if the sum of 5^n/(3^n + 4^n)5n3n+4n5^n/(3^n + 4^n) from n=0 to infinity converges?