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{{Redirect|Abel transformation|another transformation|Abel transform}}
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In [[mathematics]], '''summation by parts''' transforms the [[summation]] of products of [[sequences]] into other summations, often simplifying the computation or (especially) estimation of certain types of sums. The summation by parts formula is sometimes called '''[[Niels Henrik Abel|Abel's]] lemma''' or '''Abel transformation'''.
 
==Statement==
Suppose <math>\{f_k\}</math> and <math>\{g_k\}</math> are two [[sequence]]s. Then,
:<math>\sum_{k=m}^n f_k(g_{k+1}-g_k) = \left[f_{n+1}g_{n+1} - f_m g_m\right] - \sum_{k=m}^n g_{k+1}(f_{k+1}- f_k).</math>
 
Using the [[forward difference operator]] <math>\Delta</math>, it can be stated more succinctly as
 
:<math>\sum_{k=m}^n f_k\Delta g_k = \left[f_{n+1} g_{n+1} - f_m g_m\right] - \sum_{k=m}^n g_{k+1}\Delta f_k,</math>
 
Note that summation by parts is an analogue to the [[integration by parts]] formula,
:<math>\int f\,dg = f g - \int g\,df.</math>
 
Note also that although applications almost always deal with convergence of sequences, the statement is purely algebraic and will work in any [[Field (mathematics)|field]]. It will also work when one sequence is in a [[vector space]], and the other is in the relevant field of scalars.
 
==Newton series==
The formula is sometimes given in one of these - slightly different - forms
 
:<math>\begin{align}
\sum_{k=0}^n f_k g_k &= f_0 \sum_{k=0}^n g_k+ \sum_{j=0}^{n-1} (f_{j+1}-f_j) \sum_{k=j+1}^n g_k\\
&= f_n \sum_{k=0}^n g_k - \sum_{j=0}^{n-1} \left( f_{j+1}- f_j\right) \sum_{k=0}^j g_k,
\end{align}</math>
 
which represent a special cases (<math>M=1</math>) of the more general rule
 
:<math>\begin{align}\sum_{k=0}^n f_k g_k &= \sum_{i=0}^{M-1} f_0^{(i)} G_{i}^{(i+1)}+ \sum_{j=0}^{n-M} f^{(M)}_{j} G_{j+M}^{(M)}=\\
&= \sum_{i=0}^{M-1} \left( -1 \right)^i f_{n-i}^{(i)} \tilde{G}_{n-i}^{(i+1)}+ \left( -1 \right) ^{M} \sum_{j=0}^{n-M} f_j^{(M)} \tilde{G}_j^{(M)};\end{align}</math>
 
both result from iterated application of the initial formula. The auxiliary quantities are [[Newton series]]:
 
:<math>f_j^{(M)}:= \sum_{k=0}^M \left(-1 \right)^{M-k} {M \choose k} f_{j+k}</math>
and  
:<math>G_j^{(M)}:= \sum_{k=j}^n {k-j+M-1 \choose M-1} g_k,</math>
:<math>\tilde{G}_j^{(M)}:= \sum_{k=0}^j {j-k+M-1 \choose M-1} g_k.</math>
 
A remarkable, particular (<math>M=n+1</math>) result is the noteworthy identity
:<math>\sum_{k=0}^n f_k g_k = \sum_{i=0}^n f_0^{(i)} G_i^{(i+1)} = \sum_{i=0}^n (-1)^i f_{n-i}^{(i)} \tilde{G}_{n-i}^{(i+1)}.</math>
 
Here, <math>{n \choose k}</math> is the [[binomial coefficient]].
 
==Method==
 
For two given sequences <math>(a_n) \,</math> and <math>(b_n) \,</math>, with <math>n \in \N</math>, one wants to study the sum of the following series:<br>
<math>S_N = \sum_{n=0}^N a_n b_n</math>
 
If we define <math>B_n = \sum_{k=0}^n b_k,</math>&nbsp;
then for every <math>n>0, \,</math>&nbsp; <math>b_n = B_n - B_{n-1} \,</math>&nbsp; and
 
:<math>S_N = a_0 b_0 + \sum_{n=1}^N a_n (B_n - B_{n-1}),</math>
 
:<math>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}).</math>
 
Finally&nbsp; <math>S_N = a_N B_N - \sum_{n=0}^{N-1} B_n (a_{n+1} - a_n).</math>
 
This process, called an Abel transformation, can be used to prove several criteria of convergence for <math>S_N \,</math> .
 
==Similarity with an integration by parts==
 
The formula for an integration by parts is <math>\int_a^b f(x) g'(x)\,dx = \left[ f(x) g(x) \right]_{a}^{b} - \int_a^b  f'(x) g(x)\,dx</math><br>
Beside the [[boundary conditions]], we notice that the first integral contains two multiplied functions, one which is integrated in the final integral ( <math>g' \,</math> becomes <math>g \,</math> ) and one which is differentiated ( <math>f \,</math> becomes <math>f' \,</math> ).
 
The process of the ''Abel transformation'' is similar, since one of the two initial sequences is summed ( <math>b_n \,</math> becomes <math>B_n \,</math> ) and the other one is differenced ( <math>a_n \,</math> becomes <math>a_{n+1} - a_n \,</math> ).
 
==Applications==
 
* Summation by parts is frequently used to prove [[Abel's theorem]].
 
* If <math>\sum b_n</math> is a [[convergent series]], and <math>a_n</math> a [[monotone sequence]] decreasing to zero, then <math>S_N = \sum_{n=0}^N a_n b_n</math> remains a convergent series.
 
The [[Cauchy criterion]] gives <math>S_M - S_N = a_M B_M - a_N B_N + \sum_{n=N}^{M-1} B_n (a_{n+1} - a_n) </math>.
 
As <math>\sum b_n</math> is convergent, <math>B_N</math> is bounded independently of <math>N</math>, say by <math>B</math>. As <math>a_n</math> go to zero, so go the first two terms. The remaining sum is bounded by
 
: <math>\sum_{n=N}^{M-1} |B_n| |a_{n+1}-a_n| \le B \sum_{n=N}^{M-1} |a_{n+1}-a_n| = B(a_N - a_M)</math>
 
by the monotonicity of <math>a_n</math>, and also goes to zero as <math>N \to \infty</math>.
 
* Using the same proof as above, one shows that
# if the partial sums <math>B_N</math> remain [[bounded]] independently of <math>N</math> ;
# if <math>\sum_{n=0}^\infty |a_{n+1} - a_n| < \infty</math> (so that the sum <math>\sum_{n=N}^{M-1} |a_{n+1}-a_n|</math> goes to zero as <math>N</math> goes to infinity) ; and
# if <math>\lim a_n = 0</math>
then <math>S_N = \sum_{n=0}^N a_n b_n</math> is a convergent series.
 
In both cases, the sum of the series verifies:
<math> |S| = \left|\sum_{n=0}^\infty a_n b_n \right| \le B \sum_{n=0}^\infty |a_{n+1}-a_n|</math>
 
==See also==
*[[Convergent series]]
*[[Divergent series]]
*[[Integration by parts]]
*[[Cesàro summation]]
*[[Abel's theorem]]
*[[Abel's summation formula|Abel sum formula]]
 
==References==
*{{planetmathref|id=3843|title=Abel's lemma}}
 
[[Category:Summability methods]]
[[Category:Real analysis]]
[[Category:Lemmas]]

Latest revision as of 11:25, 14 December 2014

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