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.

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 ight] - 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_kDelta g_k = left [f_{n+1} g_{n+1} - f_m g_m ight] - 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.

Newton series

The formula is sometimes stated in the slightly different form

:$$sum_{k=0}^n f_k g_k= f_n sum_{k=0}^n g_k - sum_{j=0}^{n-1} left( f_{j+1}- f_j ight) sum_{k=0}^j g_k,

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

:$$sum_{k=0}^n f_k g_k= sum_{i=0}^{M-1} left( -1 ight)^i f_{n-i}^{(i)} G_{n-i}^{(i+1)}+ left( -1 ight) ^{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 ight)^{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.

The initial equation may be stated alternatively as:$$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.

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 , .

imilarity with an integration by parts

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

Applications

Let's consider that $$a_N b_N ightarrow 0, otherwise it is obvious that $$(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|

ee also

*Convergent series
*Divergent series
*Integration by parts
*Abel's theorem
*Abel transform

References

*planetmathref|id=3843|title=Abel's lemma