Euler-Maclaurin formula

In mathematics, the Euler-Maclaurin formula provides a powerful connection

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between integrals (see calculus) and sums. It can be used to approximate

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integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. The formula was

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discovered independently by Leonhard Euler and Colin Maclaurin

Related Topics:
Leonhard Euler - Colin Maclaurin

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around 1735. Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals.

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If n is a natural number and f(x) is a smooth (meaning: sufficiently often differentiable) function defined for all real numbers x between 0 and n, then the integral

Related Topics:
Natural number - Differentiable - Function - Real number

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:I=int_0^n f(x),dx

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can be approximated by the sum

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:

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S=rac{fleft( 0 ight) }{2}+fleft( 1 ight) +cdots+fleft( n-1 ight) +

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rac{fleft( n ight) }{2}

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We can use two expressions for S :

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:S=-rac{fleft( 0 ight) +fleft( n ight) }{2}+sum_{k=0}^{n}fleft(

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k ight)

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or

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:S=rac{fleft( 0 ight) +fleft( n ight) }{2}+sum_{k=1}^{n-1}fleft(

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k ight)

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(see trapezoidal rule). The Euler-Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives f(k) at the end points of the interval 0 and n. For any natural number p, we have

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:S-I=sum_{k=1}^prac{B_{2k}}{(2k)!}left(f^{(2k-1)}(n)-f^{(2k-1)}(0) ight)+R

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where, B2 = 1/6, B4 = −1/30, B6 = 1/42, B8 = −1/30, ... are the Bernoulli numbers.

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R is an error term which is normally small if p is large enough and can be estimated as

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:left|R ight|leqrac{2}{(2pi)^{2p}}int_0^nleft|f^{(2p+1)}(x) ight|,dx.

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By employing the substitution rule, one can adapt this formula also to functions f which are defined on some other interval of the real line.

Related Topics:
Substitution rule - Interval

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