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:

1. Approximate the sum of the series by integral

2. Induce the error of each interval [i, i + 1] using previously done approximation

3. 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!