Euler Maclaurin Formula

One of the basic concepts of calculus is the correspondence between sums and integrals, which is easily evaluated with the help of Faulhaber’s formula. The Euler–Maclaurin formula is considered as a powerful connection between integrals. It is mostly used to approximate integrals by finite sums, or conversely to calculate or evaluate finite sums and infinite series using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from the formula, and Faulhaber’s formula for the sum of powers is an immediate consequence.

\(\begin{array}{l} \sum_{k=p}^{m-1}\phi (k)=\int_{p}^{m}\phi (t)dt+\sum_{v=1}^{n-1}\frac{B_{v}}{v!}(\phi^{v-1}m-\phi^{v-1}p)+R_{n}\end{array} \)


\(\begin{array}{l}B_{v}\end{array} \)
=Bernoulli numbers
\(\begin{array}{l}R_{n}\end{array} \)