The Euler–Maclaurin formula in mathematics provides a powerful relationship between sums and integrals. This is also good to measure the integrals approximation by finite sums or you could evaluate the sum of a finite series or infinite series conversely with the helps of integral or calculus. For example, this is possible to derive a plenty of asymptotic expansions with the help of the Euler–Maclaurin formula and the sum of powers is an immediate consequence here.

\[\large \sum_{k=p}^{m-1}\phi (k)=\int_{p}^{m}\phi (t)dt+\sum_{v=1}^{n-1}\frac{B_{v}}{v!}(\phi^{v-1}m-\phi^{v-1}p)+R_{n}\]

Where,

B_{v}=Bernoulli numbers

R_{n}=remainder

The formula was first discovered in 1735 independently by Leonhard Euler and Colin Maclaurin. Euler was working on slowly converging infinite series that can be further used to calculate integrals. Then he derived one formula to provide relationship among sum or differences of integrals in terms of higher derivatives.

The formula was derived using repeated integrations by parts to the successive intervals for integers. This popular derivation is largely based on periodic Bernoulli functions, that are generally defined in terms of Bernoulli polynomials. This formula is suitable to find detailed error analysis too in numerical quadrature. This is based on the superior performance of trapezoid rule as per smooth periodic function and they are needed on certain extrapolation methods too.

The results calculated from the Euler–Maclaurin formula are highly accurate based on discrete cosine transformation. The technique is also named as the periodic transformations. Hence, this is clear that Euler–Maclaurin formula is powerful method for approximation related to the series summation to the continuous integral of a function. To understand the Euler–Maclaurin formula deeply, you should understand the Bernoulli polynomials first and move ahead.