Euler-Maclaurin Summation Formula

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Theorem

Let $f$ be a real function which is appropriately differentiable and integrable.


Then:

\(\displaystyle \sum_{k \mathop = 1}^{n - 1} f \left({k}\right)\) \(=\) \(\displaystyle \int_0^n f \left({x}\right) \rd x - \frac {f \left({0}\right) + f \left({n}\right)} 2 + \sum_{k \mathop = 1}^\infty \frac {B_{2 k} } {\left({2 k}\right)!} \left({f^{\left({2 k - 1}\right)} \left({n}\right) - f^{\left({2 k - 1}\right)} \left({0}\right)}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \int_0^n f \left({x}\right) \rd x - \frac 1 2 \left({f \left({n}\right) + f \left({0}\right)}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac 1 {12} \left({F' \left({n}\right) - F' \left({0}\right)}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(\) \(\, \displaystyle - \, \) \(\displaystyle \frac 1 {720} \left({F''' \left({n}\right) - F''' \left({0}\right)}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac 1 {30 \, 240} \left({F^{\left({5}\right)} \left({n}\right) - F^{\left({5}\right)} \left({0}\right)}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac 1 {1 \, 209 \, 600} \left({F^{\left({7}\right)} \left({n}\right) - F^{\left({7}\right)} \left({0}\right)}\right)\) $\quad$ $\quad$

where:

$f^{\left({k}\right)}$ denotes the $k$th derivative of $f$
$B_n$ denotes the $n$th Bernoulli number.


Proof


Also see


Source of Name

This entry was named for Leonhard Paul Euler and Colin Maclaurin.


Sources