Given two positive sums: $\sum_{n=1}^{\infty} a_n \ $ and $ \ \sum_{n=1}^{\infty} b_n $.
$ \ \ \exists N: \forall n \geq N: \ \dfrac{a_{n+1}}{a_n} \leq \dfrac{b_{n+1}}{b_n}$
Prove: $\sum {b_n} < \infty \Longrightarrow \sum a_n < \infty $
I tried to use the ratio test, but I can't assume that $ \dfrac{b_{n+1}}{b_n} \leq b_n$.
Even though I know that $\lim _{n \to \infty} b_n = 0$, that's not enough to determine something about $ \dfrac{b_{n+1}}{b_n}$.
Especially because the Ratio test is not of the form of "if and only if", so How can I use the fact that $ \dfrac{a_{n+1}}{a_n} \leq \dfrac{b_{n+1}}{b_n}$ in order to determine a fact about $\sum {a_n} $ convergence?
Best Answer
Note that $\left\{\dfrac{a_n}{b_n}\right\}_{n\in \Bbb N}$ is a decreasing sequence of positives numbers, hence $\left\{\dfrac{a_n}{b_n}\right\}_{n\in \Bbb N}$ is bounded.
Therefore, there is an $M>0$ with $a_n\leq M b_n$ for all $n\in \Bbb N$.
Hence $$\sum_{n=1}^{\infty}a_n\leq M \sum_{n=1}^{\infty}b_n<\infty\implies \sum_{n=1}^{\infty}a_n<\infty$$