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# 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*

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