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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 $$\int_{n=1}^{\infty}f(x)d\left \lfloor {x}\right \rfloor$$

Thanks for hints and explanations or possibly links to places where I could learn about this.

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Ok, thanks for your comments about Stieltjes integral.

So it's just like Riemann integral, except that summation points are chewed through some function (in this case floor function).

$$\int_{n=1}^{\infty}f(x)d g(x)=\sum_{i=1...\infty}f(c_i \in (x_i,x_{i+1}))(g(x_{i+1})-g(x_i))$$

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.

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