Editing Summation (section)

==Approximation by definite integrals==
Many such approximations can be obtained by the following connection between sums and [[integral]]s, which holds for any [[monotonic function|increasing]] function ''f'':

:<math>\int_{s=a-1}^{b} f(s)\ ds \le \sum_{i=a}^{b} f(i) \le \int_{s=a}^{b+1} f(s)\ ds.</math>

and for any [[monotonic function|decreasing]] function ''f'':

:<math>\int_{s=a}^{b+1} f(s)\ ds \le \sum_{i=a}^{b} f(i) \le \int_{s=a-1}^{b} f(s)\ ds.</math>

For more general approximations, see the [[Euler–Maclaurin formula]].

For summations in which the summand is given (or can be interpolated) by an [[Riemann integral|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

:<math>\frac{b-a}{n}\sum_{i=0}^{n-1} f\left(a+i\frac{b-a}n\right) \approx \int_a^b f(x)\ dx,</math>

since the right-hand side is by definition the limit for <math>n\to\infty</math> 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.