If $\sum {p_n}$ is a convergent series, does ${p_{n+1}}/{p_n} \geq {q_{n+1}}/{q_n}$ for $n \geq M$ establish convergence for $\sum q_n$?

Question

Question in title. $p_n$ and $q_n$ are positive. I was thinking that in cases where ratio test is conclusive for $p_n$ this result follows but what about cases where ratio test is inconclusive?

Answer 1

Yes to the title, since we have $$ 0 < q_{n+1} = q_1\cdot\frac{q_{n+1}}{q_1} < q_1\prod_1^n\frac{q_{j+1}}{q_j} < q_1\prod_1^n\frac{p_{j+1}}{p_j} = \frac{q_1}{p_1}p_{n+1}$$ for all $n$ and hence the comparison test gives the convergence of $\sum q_n$.

Ratio test is really useless and I dont understand why people put so much attention to it (it is just a special case where you compare with a geometric series under favourable circumstances). You should almost always use the comparison test.