Summation by parts

First, the result, which is not so much a result as it is basic symbol manipulation:

Let {a_n} and {b_n} be infinite real sequences and let {s_n} be the partial sums of {a_n}. That is,



Then



Again, to "prove" this, one merely needs to appropriately combine terms so, by itself, it's not particularly significant. It does, however, allow one to come up with a couple more convergence rules for series that converge conditionally (that is, Σa_n converges, but not Σ|a_n|.

The first is Dirichlet's Test, which I mentioned a few days ago as a more general form of Leibniz's alternating series test. Another is Abel's Test: If Σa_n converges and {b_n} is a bounded monotone sequence, then Σa_nb_n converges. Both follow from putting constraints on the series Σa_n and sequence {b_n} and then taking the limit of the summation by parts formula as n goes to infinity.

That's all well and good (and, frankly, straightforward enough to be left as an "exercise to the reader"), but what I like is that summation by parts gives a glimpse into integration by parts, which is often a very spooky result to Freshman Calculus students (it certainly was for me).

The similarity in form is pretty obvious if you re-label the differences in the partial sums in the original formulation: db_k = b_k+1 - b_k and ds_k = s_k - s_k-1 = a_kThen summation by parts is:




There's still some non-trivial work to do, but it's not terribly hard to see how taking this limit winds up giving:



Not a proof, by any means, but it does give some intuition into why it works.