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Maclaurin-Cauchy Theorem

If f(x) is positive and decreases to 0, then an Euler constant

gamma_f=lim_(n->infty)[sum_(k=1)^nf(k)-int_1^nf(x)dx]

can be defined.

For example, if f(x)=1/x, then

gamma=lim_(n->infty)(sum_(k=1)^n1/k-int_1^n(dx)/x)=lim_(n->infty)(sum_(k=1)^n1/k-lnn),

which is just the usual Euler-Mascheroni constant.

See also

Euler-Maclaurin Integration Formulas, Euler-Mascheroni Constant, Numerical Differentiation

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Cite this as:

Weisstein, Eric W. "Maclaurin-Cauchy Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Maclaurin-CauchyTheorem.html

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