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1 [made Community Wiki]

Given a function $f$ on the real line, let's compute the function $\Sigma f$, taking $n \mapsto f(1) +\ldots + f(n)$. Well, $\Sigma = 1/\Delta$, where $\Delta$ is the differencing operator $shift - 1$. And the shift operator is the exponential of the differentiation operator (this being, essentially, Taylor's theorem). Hence $$\Sigma = \frac{1}{e^D - 1} = \frac{1}{D} \frac{D}{e^D-1}$$ Using L'Hopital's rule on the latter as $D\to 0$, whatever THAT means, we see the limit is $1$. So expand in a power series: $$\frac{1}{D} \frac{D}{e^D-1} = \frac{1}{D} (1 + \text{power series in D})$$ The first term is $1/D$, which is of course $\int$.

No surprise: $\Sigma = \int +$ correction terms. What the above suggests is that those correction terms come from the Taylor expansion of $\frac{D}{e^D - 1}$. This leads to the Euler summation formula (and eventually, to Hirzebruch-Riemann-Roch).

I learned this from "Concrete Mathematics", where I recall this joke being attributed to Laguerre. Part of why it is a joke is that the Euler summation formula has an error term, that can't be neglected for most functions, e.g. $\ln(x)$ which one wants to sum up to compute $\ln(n!)$. It can be neglected for polynomials times exponentials.