Math Euler Maclaurin Formula with Problem Solution & Solved Example More Videos 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. Related posts: What is Integration? List of Integration by Parts Formulas Integrals Maths Formulas for Class 12 Chapter 7 Infinite Geometric Series Formula, Hyper Geometric Sequence Distribution List of Basic Maths Formulas for Class 5 to 12 Applications of Integrals Maths Formulas for Class 12 Chapter 8 Maths Formulas for Class 12 Sequences and Series Formulas for Class 11 Maths Chapter 9 Maths Formulas For Class 11 Definite Integral Formula Polynomials Formulas for Class 9 Maths Chapter 2 List of Maths Formulas for Class 10th CBSE Statistics Formulas for Class 10 Maths Chapter 14 Polynomials Formulas for Class 10 Maths Chapter 2 What is Polynomial? List of Basic Polynomial Formula U Substitution Formula – Problem Solution with Solved Example Prime Number Formula with Problem Solution & Solved Example Relations and Functions Formulas for Class 11 Maths Chapter 2 Three Dimensional Geometry Maths Formulas for Class 12 Chapter 11 Right Angle Formula| Half-Angle, Double Angle, Multiple Difference of Squares Formula | Regression Sum of Squares Formula Related Topics: Up Next Prime Number Formula with Problem Solution & Solved Example Don't Miss Frequency Distribution Formula with Problem Solution & Solved Example Continue Reading