|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
summation by parts
|
(Theorem)
|
|
|
The following corollaries apply Abel's lemma to allow estimation of certain bounded sums:
Corollary 1 (Summation by parts)
Let $\{a_i\},\{b_i\}$ be sequences of complex numbers. Suppose the partial sums of the $a_i$ are bounded in magnitude by $h$ , that $\sum_0^{\infty} |b_i-b_{i+1}|$ converges, and that $\lim_{i\to\infty} b_i=0$ . Then $\sum_0^{\infty} a_i b_i$ converges, and $$ \left|\sum_0^{\infty}a_i b_i\right|\leq h\sum_0^{\infty}|b_i-b_{i+1}
$$
Proof. By Abel's lemma, $$ \sum_{i=0}^N a_ib_i = \sum_{i=0}^{N-1} A_i(b_i-b_{i+1}) + A_Nb_ $$ so that
The condition that the $b_i\to 0$ is easily seen to imply that the sequence $\left\lvert \sum_{i=0}^N a_ib_i\right\rvert$ is Cauchy hence convergent, so that $$ \left\lvert \sum_{i=0}^\infty a_ib_i\right\rvert \leq h\sum_{i=0}^{\infty} \left\lvert b_i-b_{i+1}\right\rver $$ since $b_N\to 0$ .
Corollary 2 (Summation by parts for real sequences)
Let $\{a_i\}$ be a sequence of complex numbers. Suppose the partial sums are bounded in magnitude by $h$ . Let $\{b_i\}$ be a sequence of decreasing positive real numbers such that $\lim_{i\to\infty} b_i=0$ . Then $\sum_1^{\infty} a_ib_i$ converges, and $|\sum_1^{\infty} a_ib_i|\leq hb_1$ .
Proof. This follows immediately from the above, since $|b_i-b_{i+1}|=b_i-b_{i+1}$ .
|
"summation by parts" is owned by rm50. [ full author list (2) ]
|
|
(view preamble | get metadata)
| Other names: |
partial summation |
This object's parent.
|
|
Cross-references: positive, decreasing, real, convergent, imply, proof, converges, bounded, partial sums, complex numbers, sequences, bounded sums
There are 2 references to this entry.
This is version 5 of summation by parts, born on 2006-12-17, modified 2008-04-30.
Object id is 8632, canonical name is PartialSummation.
Accessed 2490 times total.
Classification:
| AMS MSC: | 40D05 (Sequences, series, summability :: Direct theorems on summability :: General theorems) | | | 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
