## You are here

HomeAbel's lemma

## Primary tabs

# Abel’s lemma

Theorem 1 Let $\{a_{i}\}_{{i=0}}^{N}$ and $\{b_{i}\}_{{i=0}}^{N}$ be sequences of real (or complex) numbers with $N\geq 0$. For $n=0,\ldots,N$, let $A_{n}$ be the partial sum $A_{n}=\sum_{{i=0}}^{n}a_{i}$. Then

$\sum_{{i=0}}^{N}a_{i}b_{i}=\sum_{{i=0}}^{{N-1}}A_{i}(b_{i}-b_{{i+1}})+A_{N}b_{% N}.$ |

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.

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 $0\leq M\leq N$. For $n=M,\ldots,N$, let $A_{n}$ be the partial sum $A_{n}=\sum_{{i=M}}^{n}a_{i}$. Then

$\sum_{{i=M}}^{N}a_{i}b_{i}=\sum_{{i=M}}^{{N-1}}A_{i}(b_{i}-b_{{i+1}})+A_{N}b_{% N}.$ |

*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 $\{b_{i}\}_{{i=0}}^{N}$.
$\Box$

# References

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

## Mathematics Subject Classification

40A05*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith

## Attached Articles

## Corrections

one term missing by paolini ✓

one term missing by paolini ✘

synonyms by pahio ✓

Synonyms by pahio ✓