The sum of a series may not be easily found, and it is desirable to be able to find upper and lower limits for its value. If series consists of terms that are decreasing or decreasing then it may be possible to easily find a bound by integration.

To find the bounds fornote thatis an increasing function. To find a lower bound divide the interval of integrationintointervalsThe minimum value ofon each intervalisThe area of the (k-1)th rectangle is {1 over n}e^{-{n over k}} .

Hence

To find an upper bound for the integral note that the maximum value ofon each intervalisThe area of theth rectangle, is

Hence

Hence we can write