Ratio comparison test ($a_{n+1}/a_n\le b_{n+1}/b_n$)

For positive sequences $(a_n),(b_n)$, suppose $a_{n+1}/a_n\le b_{n+1}/b_n$ eventually. Then $a_n/b_n$ is eventually nonincreasing, so $a_n\le C{b}_n$ beyond a fixed index. The comparison test for series proves convergence of $\sum a_n$ whenever $\sum b_n$ converges. Reversing the ratio inequality gives an eventual lower bound $a_n\ge c{b}_n$ and hence divergence when the positive comparison series diverges. The initial finite set of terms does not affect either conclusion.