Starting in page 5, the authors discuss how the "naive" approach can be improved upon for this particular example by bounding the error. My idea is to (i) verify that this series conforms to the assumptions (ii) given (i) holds, show that the adaptive (bounding pair) approach correctly recovers the right $\kappa$ (in their notation), i.e. the minimum number of iterations to achieve error below a certain threshold.
https://digital.library.adelaide.edu.au/dspace/bitstream/2440/60405/2/hdl_60405.pdf
Starting in page 5, the authors discuss how the "naive" approach can be improved upon for this particular example by bounding the error. My idea is to (i) verify that this series conforms to the assumptions (ii) given (i) holds, show that the adaptive (bounding pair) approach correctly recovers the right $\kappa$ (in their notation), i.e. the minimum number of iterations to achieve error below a certain threshold.