The Euler-Maclaurin formula is a mathematical tool that connects discrete sums and continuous integrals. It provides a way to approximate the sum of a function's values at discrete points by relating it to the integral of that function over an interval. This formula is particularly useful in numerical analysis and helps improve the accuracy of approximations. The formula incorporates corrections based on the function's derivatives at the endpoints of the interval, allowing for better estimates. It is named after mathematicians Leonhard Euler and Carl Friedrich Gauss, who contributed to its development. The Euler-Maclaurin formula is widely used in various fields, including physics and engineering.