= Ratio comparison test
{title2=$a_{n+1}/a_n\leq b_{n+1}/b_n$}
For positive <sequences> $(a_n),(b_n)$, suppose $a_{n+1}/a_n\leq b_{n+1}/b_n$ eventually. Then $a_n/b_n$ is eventually nonincreasing, so $a_n\leq Cb_n$ 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 $a_n\geq cb_n$ and hence divergence when the positive comparison <series> diverges. The initial finite set of terms does not affect either conclusion.
Back to article page
