We can estimate the value of the sum of a series of termsby representing the sum by a sequence of rectangles of base 1 and heightforThis can also be estimated using the integral

ThenThe integral is an underestimate for the sum of the series since since- we can add a sequence of these expressions to get

Similarly We can estimate the value of the sum of the seriesby representing the sum by a sequence of rectangles of base 1 and heightforThis can also be estimated using the integral

ThenThe integral is an overestimate for the sum of the series since since- we can add a sequence of these expressions to get

This is illustrated in the diagram.

Hence

In general ifforthen

and ifforthen