Recently I found an interesting way to approximate integrable series that is monotone in derivation of all order, for example, 
. I'd call it 'Recursive Integration'.
Do these things in step:
Approximate the sum of the series by integral
Induce the error of each interval [i, i + 1] using previously done approximation
Let the error be a new series, approximate this series by returning to step 1, and add it as an error to the original approximation.
Please comment if you tried this method. Thanks!
