Summation by parts

The summation by parts is the equivalent for the series of the Intégration by parts. It is also called transformation of Abel or summation of Abel .

Method

Are two continuations $(a_n) \,$ and $(b_n) \,$ , with $n \ in \ N$ . The following series is considered:
$S_N = \ sum_ {n=0} ^N a_n b_n$

If one poses $B_n = \ sum_ {k=0} ^n b_k$ ,
then for all n>0, $b_n = B_n - B_ {n-1} \,$

$S_N = a_0 b_0 + \ sum_ {n=1} ^N a_n (B_n - B_ {n-1})$
$S_N = a_0 b_0 - a_1 B_0 + a_N B_N + \ sum_ {n=1} ^ {N-1} B_n (a_n - a_ {n+1})$
One obtains finally the following equality: $S_N = a_N B_N - \ sum_ {n=0} ^ {N-1} B_n (a_ {n+1} - a_n)$

This operation, which transforms the expression of the series to be studied, is useful to prove certain criteria of convergence of $S_N \,$ .

Similarity with integration by parts

The formula of integration is written: $\ int_a^b F (X) g' (X) \, dx = \ left F (X) G (X) \ right_ {has} ^ {B} - \ int_a^b f' (X) G (X) \, dx$
If one leaves side the boundary conditions, one realizes that integration by parts consists in integrating one of the two functions present in the initial integral ( $g' \,$ becomes $g \,$ ) and to derive the other ( $f \,$ becomes $f' \,$ ).

The summation by parts consists of a similar operation in the discrete field, since one of the two series is summoned ( $b_n \,$ becomes $B_n \,$ ) and the other is differentiated ( $a_n \,$ becomes $a_ {n+1} - a_n \,$ ).

Applications

One places oneself thereafter if $a_N b_N \ rightarrow 0$ , because if not one knows that $(S_N) \,$ is trivialement divergent.

If $(B_n) \,$ is limited by a reality M and that $\ sum_ {k=0} ^N (a_ {n+1} - a_n)$ is an absolutely convergent series , then the series $(S_N) \,$ is convergent.

$|S_N| \ it |a_N B_N| + \ sum_ {n=0} ^ {N-1} |B_n| |a_ {n+1} - a_n|$

The sum of the series checks the inequality in addition: $S = \ sum_ {n=0} ^ \ infty a_n b_n \ the M \ sum_ {n=0} ^ \ infty |a_ {n+1} - a_n|$

Examples

$a_n = \ frac {1} {n+1}$ and $b_n = (- 1) ^n \,$
$|B_n| \ the 1$ and $|a_ {n+1} - a_n| = \ frac {1} {(n+1) (n+2)} \ the \ frac {1} {n^2}$
On knows that the series $\ sum_0^ \ infty \ frac {1} {n^2}$ converges (see Fonction zeta of Riemann), therefore the conditions mentioned above are all réunies.
$S = 1 - \ frac {1} {2} + \ frac {1} {3} - \ frac {1} {4} +…$ converge.
NB: This example can also be proven thanks to the Critère of convergence of the alternate series.
$a_n = \ frac {1} {N}$ and $b_n = \ sin (N) \,$
(We define here the sum only starting from the row n=1 instead of n=0, but that does not affect of anything the existence limit of the series.)
Comme previously $\ sum_ {n=1} ^ \ infty (\ frac {1} {n+1} - \ frac {1} {N})$ converges absolutely, and $\ sum_ {k=1} ^n \ sin (K)$ is limited according to the expression of the Noyau of Dirichlet.
Consequently $\ sum_ {n=1} ^ \ infty \ frac {\ sin (N)}{N}$ converges.
the summation by parts is useful in the proof of the theorem of Abel.

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