Given that f is a continuous function and [x] is the biggest integer which is smaller than x,
I want to integrate f(x)d[x] from 0 to n, where n is a natural number.
I know the definition of the Riemann-Stieltjes integral,
but I can't actually use the definition to calculate the examples...
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Isn't this just $f(1)+f(2)+\cdots+f(n)$? – MPW Jun 03 '15 at 16:25
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MPW is correct. In a bit more detail, $[x]$ changes exactly at the integers. At each integer it jumps up by $1$. So an integral over an interval with respect to $[x]$ is a sum of the values of the integrand at the integers contained in the interval, each multiplied by the jump amount (i.e. by $1$). – Ian Jun 03 '15 at 16:29