Summation by parts

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 Abel's lemma or Abel transformation.

Statement

Suppose \( \{f_k\} \) and \( \{g_k\} \) are two sequences. Then,

\( \sum_{k=m}^n f_k(g_{k+1}-g_k) = \left[f_{n+1}g_{n+1} - f_m g_m\right] - \sum_{k=m}^n g_{k+1}(f_{k+1}- f_k). \)

Using the forward difference operator \Delta, it can be stated more succinctly as

\( \sum_{k=m}^n f_k\Delta g_k = \left[f_{n+1} g_{n+1} - f_m g_m\right] - \sum_{k=m}^n g_{k+1}\Delta f_k, \)

Note that summation by parts is an analogue to the integration by parts formula,

\( \int f\,dg = f g - \int g\,df. \)

Note also that although applications almost always deal with convergence of sequences, the statement is purely algebraic and will work in any 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

\( \begin{align}\sum_{k=0}^n f_k g_k &= f_0 \sum_{k=0}^n g_k+ \sum_{j=0}^{n-1} (f_{j+1}-f_j) \sum_{k=j+1}^n g_k=\\ &= f_n \sum_{k=0}^n g_k - \sum_{j=0}^{n-1} \left( f_{j+1}- f_j\right) \sum_{k=0}^j g_k, \end{align} \)

which represent a special cases (M=1) of the more general rule

\( \begin{align}\sum_{k=0}^n f_k g_k &= \sum_{i=0}^{M-1} f_0^{(i)} G_{i}^{(i+1)}+ \sum_{j=0}^{n-M} f^{(M)}_{j} G_{j+M}^{(M)}=\\ &= \sum_{i=0}^{M-1} \left( -1 \right)^i f_{n-i}^{(i)} \tilde{G}_{n-i}^{(i+1)}+ \left( -1 \right) ^{M} \sum_{j=0}^{n-M} f_j^{(M)} \tilde{G}_j^{(M)};\end{align} \)

both result from iterated application of the initial formula. The auxiliary quantities are Newton series:

\( f_j^{(M)}:= \sum_{k=0}^M \left(-1 \right)^{M-k} {M \choose k} f_{j+k} \)

and

\( G_j^{(M)}:= \sum_{k=j}^n {k-j+M-1 \choose M-1} g_k,
\tilde{G}_j^{(M)}:= \sum_{k=0}^j {j-k+M-1 \choose M-1} g_k. \)

A remarkable, particular (M=n+1) result is the noteworthy identity

\( \sum_{k=0}^n f_k g_k = \sum_{i=0}^n f_0^{(i)} G_i^{(i+1)} = \sum_{i=0}^n (-1)^i f_{n-i}^{(i)} \tilde{G}_{n-i}^{(i+1)}. \)

Here,\( {n \choose k} \)is the binomial coefficient.
Method

For two given sequences \( (a_n) \,\) and \( (b_n) \,, \) with \( n \in \N, \)one wants to study the sum of the following series:
\( S_N = \sum_{n=0}^N a_n b_n \)

If we define \( B_n = \sum_{k=0}^n b_k, then for every n>0, \, b_n = B_n - B_{n-1} \, \)and

\( S_N = a_0 b_0 + \sum_{n=1}^N a_n (B_n - B_{n-1}), \)

\( S_N = a_0 b_0 - a_1 B_0 + a_N B_N + \sum_{n=1}^{N-1} B_n (a_n - a_{n+1}). \)

Finally \( S_N = a_N B_N - \sum_{n=0}^{N-1} B_n (a_{n+1} - a_n). \)

This process, called an Abel transformation, can be used to prove several criteria of convergence for S_N \, .
Similarity with an integration by parts

The formula for an integration by parts is \( \int_a^b f(x) g'(x)\,dx = \left[ f(x) g(x) \right]_{a}^{b} - \int_a^b f'(x) g(x)\,dx \)
Beside the boundary conditions, we notice that the first integral contains two multiplied functions, one which is integrated in the final integral \( ( g' \, \) becomes \( g \, \) ) and one which is differentiated ( \( f \, \) becomes \( f' \, \)).

The process of the Abel transformation is similar, since one of the two initial sequences is summed \( ( b_n \, \) becomes\( B_n \,\) ) and the other one is differenced ( \( a_n \, \) becomes \( a_{n+1} - a_n \, \) ).
Applications

We suppose that a_N b_N \rightarrow 0; otherwise it is obvious that (S_N)\, is a divergent series.

If \( (B_n) \ \), is bounded by a real M and \( \sum_{n=0}^N (a_{n+1} - a_n) \)is absolutely convergent, then \( (S_N)\, \) is a convergent series.

\( |S_N| \le |a_N b_N| + \sum_{n=0}^{N-1} |B_n| |a_{n+1}-a_n| \)

And the sum of the series verifies: \( S = \sum_{n=0}^\infty a_n b_n \le M \sum_{n=0}^\infty |a_{n+1}-a_n| \)
See also

Convergent series
Divergent series
Integration by parts
Cesàro summation
Abel's theorem
Abel sum formula

References

Abel's lemma, PlanetMath.org.