# Sum as an integral

Recently I have encountered weird notation that I don't see into.

When I have some infinite sum $$\sum_{n=1}^{\infty}f(n)$$ I would rewrite it without thinking to the integral form like this $$\int_{n=1}^{\infty}f(\left \lfloor {x}\right \rfloor)dx$$ where $\left \lfloor {x}\right \rfloor$ is a floor function.

The thing I saw and have difficulty to interpret looked like this $$\int_{n=1}^{\infty}f(x)d\left \lfloor {x}\right \rfloor$$

$$\int_{n=1}^{\infty}f(x)d g(x)=\sum_{i=1...\infty}f(c_i \in (x_i,x_{i+1}))(g(x_{i+1})-g(x_i))$$
In my case the right bracket $(g(x_{i+1})-g(x_i))$ will be zero everywhere but in integers where the value will be one.