Approximation By Definite Integrals
Many such approximations can be obtained by the following connection between sums and integrals, which holds for any:
increasing function f:
decreasing function f:
For more general approximations, see the Euler–Maclaurin formula.
For summations in which the summand is given (or can be interpolated) by an integrable function of the index, the summation can be interpreted as a Riemann sum occurring in the definition of the corresponding definite integral. One can therefore expect that for instance
since the right hand side is by definition the limit for of the left hand side. However for a given summation n is fixed, and little can be said about the error in the above approximation without additional assumptions about f: it is clear that for wildly oscillating functions the Riemann sum can be arbitrarily far from the Riemann integral.
Read more about this topic: Summation
Famous quotes containing the word definite:
“Mathematics may be compared to a mill of exquisite workmanship, which grinds your stuff of any degree of fineness; but nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat- flour from peascods, so pages of formulae will not get a definite result out of loose data.”
—Thomas Henry Huxley (1825–95)