Abel transformation

2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL] summation by parts, Abel's lemma

A discrete analog of integration by parts, introduced by N.H. Abel in [Ab]. If $a_1, \ldots, a_N$, $b_1, \ldots, b_N$, are given complex numbers and we set \[ B_n = \sum_{i\leq n} b_i \] then the summation by parts is the identity \[ \sum_{k=1}^N a_k b_k = a_N B_N - \sum_{k=1}^{N-1} B_k (a_{k+1} - a_k)\, . \] Observe that, if $B_0$ is arbitrarily chosen and we modify the definition of $B_n$ by \[ B_n = B_0 + \sum_{i\leq n} b_i \] then the identity becomes \[ \sum_{k=1}^N = a_N B_N - a_1 B_0 - \sum_{k=1}^{N-1} B_k (a_{k+1} - a_k)\, , \] in analogy with the arbitrarity of an additive constant in the primitive of a function.

If $a_n\to 0$ and $\{B_n\}$ is a bounded sequence, the Abel transformation shows that $\sum_k a_k b_k$ converges if and only if $\sum_k B_k (a_{k+1}-a_k)$ converges, in which case it yields the formula \[ \sum_{k=1}^\infty a_k b_k = \sum_{k=1}^\infty B_k (a_k - a_{k+1}) - a_1 B_0\, . \] This fact can be used to prove several very useful criteria of convergence of series of numbers and functions (cf. Abel criterion). The Abel transformation of a series often yields a series with an identical sum, but with a better convergence. It is also regularly used to obtain certain estimates (cf. Abel inequality), in particular, for investigations on the rate of convergence of a series.

References

[Ab]	N.H. Abel, "Untersuchungen über die Reihe $1+ \frac{m}{x} + \frac{m\cdot (m-1)}{2\cdot 1} x^2 + \frac{m\cdot (m-1)\cdot (m-2)}{3\cdot 2\cdot 1} x^3 + \ldots$ u.s.w.", J. Reine Angew. Math. , 1 (1826) pp. 311–339
[Ca]	H. Cartan, "Elementary Theory of Analytic Functions of One or Several Complex Variable", Dover (1995).
[Ma]	A.I. Markushevich, "Theory of functions of a complex variable" , 1 , Chelsea (1977) (Translated from Russian)
[WW]	E.T. Whittaker, G.N. Watson, "A course of modern analysis" , 1–2 , Cambridge Univ. Press (1952)