Abel’s Partial Summation Formula
When we consider an infinite series we construct the sequence of partial sums of the series. This is something like the indefinite integral of the sequence of terms of the series.
What’s the analogue of differentiation? We simply take a sequence and write and for . Then we can take the sequence of partial sums
Similarly, we can take the sequence of differences of a sequence of partial sums
This behaves a lot like the Fundamental Theorem of Calculus, in that constructing the sequence of partial sums and constructing the sequence of differences invert each other.
Now how far can we push this analogy? Let’s take two sequences, and . We define the sequence of partial sums and the sequence of differences and . We calculate
This is similar to the formula for integration by parts, and is referred to as Abel’s partial summation formula. In particular, it tells us that the series converges if both the series and the sequence converge.