Prove $\sum^{\infty}_{n=1} \frac{a_{n}-a_{n-1}}{a_{n}}=\infty$ [duplicate]

Prove $$\sum^{\infty}_{n=1} \frac{a_{n}-a_{n-1}}{a_{n}}=\infty$$ Where $a_{n}$ is an increasing sequence of positive terms that goes to infinity.

I tried to approach it with $\log(a_{n})$ like a classical version of this problem but i could not show that the difference with respect to the logarithm is finite. Any help or hint will be appreciated

Solution 1:

Show that:

$$\sum_{n=N+1}^M \frac{a_n-a_{n-1}}{a_n} \geq \frac{a_M-a_N}{a_M}$$

Then pick an increasing sequence $N_i$ so that $a_{N_{i+1}}\geq 2A_{N_i}$.

Then $$\sum_{n=N_i+1}^{N_{i+1}}\frac{a_n-a_{n-1}}{a_n}\geq 1$$

Use this to deduce the series diverges.