Right Riemann sum of divergent integral diverges

Question

Suppose $f$ is continuous and decreasing with $\int_1^\infty f(x) dx = \infty$. Fix any increasing sequence $x_i \uparrow \infty$. Is it true that $\sum_{i=1}^\infty f(x_{i+1}) (x_{i+1} - x_i) = \infty$?

Answer 1

No

You can choose your $x_i$ such that the area under $y=f(x_{i+1})$ over each $$[x_i, x_{i+1} ]$$ is less than $$ 1/2^i$$

Then you have $$\sum_{i=1}^\infty f(x_{i+1}) (x_{i+1} - x_i) <1 $$