Calculus of a Single Variable, Chapter 9, 9.4, Section 9.4, Problem 12

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 sumb_n converges then suma_n converges
If suma_n diverges then sumb_n diverges
Let a_n=3^n/2^n=(3/2)^n and b_n=3^n/(2^n-1)
3^n/(2^n-1)>3^n/2^n>0 for n>=1
sum_(n=1)^oo(3/2)^n is a geometric series with ratio r=3/2>1
A geometric series with |r|>=1 diverges.
The geometric series sum_(n=1)^oo(3/2)^n diverges and so the series sum_(n=1)^oo3^n/(2^n-1) diverges as well, by the direct comparison test.