Talk:Summation by parts

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First, there is a typo in the formula for the partial sums, in the last formula of the "Methods" section, the first b_N should be a B_N. Next, I believe that the hypotheses in the "Applications" section is not sufficient for the result stated. In the current case, you can take the sequence a_n to be a constant sequence, say identically 1, and the sequence b_n to be the following: 1, -1/2, -1/2, 1/3, 1/3, 1/3, ... (there are n terms each of 1/n, with alternating signs between each group). The sequence a_nb_n approaches 0, the series (a_(n+1)-a_n) is absolutely convergent, and the partial sums B_n are uniformly bounded (they are all between 0 and 1). However, the sum a_nb_n does not converge, since it is just the sum of b_n which oscillates infinitely often between 0 and 1. The correct hypothesis for convergence should be to replace "a_nb_n approches 0", by the condition that "a_n approaches 0". This can be seen by writing the Cauchy sequence of the partial sums in the summation by parts formulation and then estimating each part.