Sum - Approximation By Definite Integrals

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.