Calculus of a Single Variable, Chapter 9, 9.4, Section 9.4, Problem 6

Direct comparison test is applicable when suma_n and sumb_n are both positive series for all n such that a_n<=b_n
If b_n converges ,then a_n converges.
If a_n diverges, then b_n diverges.
sum_(n=0)^oo4^n/(5^n+3)
Let a_n=4^n/(5^n+3) and b_n=4^n/5^n=(4/5)^n
4^n/5^n>4^n/(5^n+3)>0 for n>=1
sum_(n=0)^oo(4/5)^n is a geometric series with ratio r=4/5<1
A geometric series with ratio r , such that |r|<1 converges.
The geometric series sum_(n=0)^oo(4/5)^n converges,so the series sum_(n=0)^oo4^n/(5^n+3) converges as well , by the direct comparison test.