Summation by parts

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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 Abel's lemma or Abel transformation.

1 Definition
2 Newton series
3 Method
4 Similarity with an integration by parts
5 Applications
6 See also
7 References

[edit] Definition

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 $Δ$ , 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,

$\int f\,dg = f g - \int g\,df.$

[edit] Newton series

The formula is sometimes stated in the slightly different form

$\sum_{j=0}^n f_j g_j= f_n \sum_{k=0}^n g_k - \sum_{j=0}^{n-1} \left( f_{j+1}- f_j\right) \sum_{k=0}^j g_k$ ,

which itself is a special case ( $M = 1$ ) of this more general rule

$\sum_{j=0}^n f_j g_j= \sum_{i=0}^{M-1} \left( -1 \right)^i f_{n-i}^{(i)} G_{n-i}^{(i+1)}+ \left( -1 \right) ^{M} \sum_{j=0}^{n-M} f_j^{(M)} G_j^{(M)}$ ,

which results 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

$G_j^{(M)}= \sum_{k=0}^j {j-k+M-1 \choose M-1} g_k$ .

Here, ${n \choose k}$ is the binomial coefficient.

[edit] 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, $b_n = B_n - B_{n-1} \,$

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

[edit] 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 derivated ( $f \,$ 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 discretely derivated ( $a_n \,$ becomes $a_{n+1} - a_n \,$ ).