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The remainder in an approximation formula is the difference between the exact and the approximate values of the expression represented by the formula. A remainder can take different forms depending on the nature of the approximation formula. The task of investigating a remainder usually consists in obtaining estimates for it. For example, corresponding to the approximate formula
we have the exact equality
where the expression R is the remainder for the approximation 1.41 for the number and it is known that 0.004 < R < 0.005.
Remainders are constantly encountered in asymptotic formulas. For example, for the number π (x) of primes not exceeding x we have the asymptotic formula
where μ is any positive number less than 3/5. Here, the remainder, which is the difference between the functions π (x) an ∫x2 du/ln u for x ≥ 2, is written in the form
where the letter O indicates that the remainder does not exceed the expression
in absolute value, C being some positive constant. Remainders are found in formulas that give approximate representations of functions. For example, in the Taylor formula
the remainder Rn (x) in Lagrange’s form is
where θ is a number such that 0 < θ < 1; θ generally depends on the values of x and h. The presence of 0 in the formula for Rn(x) introduces an element of indefiniteness; such indefinite-ness is inherent in many formulas for the remainder.
Remainders also occur in quadrature formulas and interpolation formulas.