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Abel's lemma (Theorem)

Theorem 1 Let $ \{a_i\}_{i=0}^N$ and $ \{b_i\}_{i=0}^N$ be sequences of real (or complex) numbers with $ N\ge 0$. For $ n=0,\ldots, N$, let $ A_n$ be the partial sum $ A_n=\sum_{i=0}^na_i$. Then

$\displaystyle \sum_{i=0}^N a_i b_i = \sum_{i=0}^{N-1}A_i(b_i-b_{i+1})+A_N b_N.$

In the trivial case, when $ N=0$, then sum on the right hand side should be interpreted as identically zero. In other words, if the upper limit is below the lower limit, there is no summation.

An inductive proof can be found here. The result can be found in [1] (Exercise 3.3.5).

If the sequences are indexed from $ M$ to $ N$, we have the following variant:

Corollary Let $ \{a_i\}_{i=M}^N$ and $ \{b_i\}_{i=M}^N$ be sequences of real (or complex) numbers with $ 0\le M \le N$. For $ n=M,\ldots, N$, let $ A_n$ be the partial sum $ A_n=\sum_{i=M}^na_i$. Then

$\displaystyle \sum_{i=M}^N a_i b_i = \sum_{i=M}^{N-1}A_i(b_i-b_{i+1})+A_N b_N.$

Proof. By defining $ a_0=\ldots =a_{M-1}=b_0=\ldots =b_{M-1} =0$, we can apply Theorem 1 to the sequences $ \{a_i\}_{i=0}^N$ and $ \{b_i\}_{i=0}^N$. $ \Box$

Bibliography

1
R.B. Guenther, L.W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Dover Publications, 1988.



"Abel's lemma" is owned by mathcam. [ full author list (3) | owner history (1) ]
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See Also: summation by parts

Other names:  summation by parts, Abel's partial summation, Abel's identity, Abel's transformation

Attachments:
proof of Abel's lemma (by induction) (Proof) by mathcam
summation by parts (Theorem) by rm50
example of summation by parts (Example) by pahio
proof of Abel lemma (by expansion) (Proof) by perucho
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Cross-references: lower limit, upper limit, right hand side, sum, partial sum, numbers, complex, real, sequences
There are 6 references to this entry.

This is version 11 of Abel's lemma, born on 2002-12-27, modified 2007-09-11.
Object id is 3843, canonical name is AbelsLemma.
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AMS MSC: 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences)
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