The ratio test states that a series sum_(n=0)^oo a_n converges absolutely if:
lim_(n->oo) abs (a_(n+1)/a_n) < 1
Let us determine the ratio for the series:
sum_(n=0)^oo (4^n)/(3^n+5^n)x^n
abs (a_(n+1)/a_n) = (4^(n+1)/(3^(n+1)+5^(n+1))abs(x)^(n+1))/(4^n/(3^n+5^n)abs(x)^n)=4abs(x) (3^n+5^n)/(3^(n+1)+5^(n+1))=4/5abs(x) (1+(3/5)^n)/(1+(3/5)^(n+1))
Now, as 3/5 < 1 we have that:
lim_(n->oo) (3/5)^n = 0
so that:
lim_(n->oo) abs (a_(n+1)/a_n) = lim_(n->oo) 4/5abs(x) (1+(3/5)^n)/(1+(3/5)^(n+1)) = 4/5absx
We can then conclude that for:
absx <5/4 => lim_(n->oo) abs (a_(n+1)/a_n) <1 and the series is absolutely convergent
absx > 5/4 => lim_(n->oo) abs (a_(n+1)/a_n) >1 and the series is divergent,
The case where abs(x) = 5/4 is indeterminate and we have to analyze in detail.
In the case where x=+-5/4 we have:
abs(a_n) = 4^n/(3^n+5^n)(5/4)^n = 5^n/(3^n+5^n) = 1/(1+(3/5)^n)
so that:
lim_(n->oo) abs(a_n) = 1 >0
and that means that the series cannot converge.
In conclusion the series is convergent in the interval x in (-5/4,5/4) where it is also absolutely convergent.