Let $\left\{a_{n}\right\}$ be a strictly increasing sequence of positive numbers,
Do $$\sum_{n=0}^{\infty}\frac{1}{a_{n}}$$ and $$\sum_{n=0}^{\infty}\frac{a_{n-1}}{a_{n}}$$ and $$\sum_{n=0}^{\infty}e^{-a_{n}}$$ MUST converge ? I have tried ratio test,but it seems does not work. How to test the series converge or not?