A problem on sequence and series Suppose we have a decreasing sequence $\{x_n\}$ which converges to $0$. 
Then is it true that the sum $$\sum_{n=1}^{\infty}\frac{x_n-x_{n-1}}{x_n}$$ diverges ?

Here is a potentially useful trick. Since the $x_n$ are decreasing, you can take a chunk like:
$$\frac{x_{n+1} - x_n}{x_{n+1}} + \frac{x_{n+2} - x_{n+1}}{x_{n+2}} + \cdots + \frac{x_{m+n} - x_{m+n-1}}{x_{m+n}}$$
and bound each of the denominators by $x_n$, so 
\begin{align*}&\frac{x_{n+1} - x_n}{x_{n+1}} + \frac{x_{n+2} - x_{n+1}}{x_{n+2}} + \cdots + \frac{x_{m+n} - x_{m+n-1}}{x_{m+n}} \\ \leq &\frac{x_{n+1} - x_n}{x_n} + \frac{x_{n+2} - x_{n+1}}{x_n} + \cdots + \frac{x_{m+n} - x_{m+n-1}}{x_n}\end{align*}
Note something nice happens in the numerators.
