Summation by parts

In mathematics, the abelian partial summation ( to NH Abel ) a certain transformation of a sum of products of every two numbers.

Statement

There are a natural number, and real numbers. Then we have

With

The statement has a certain formal similarity for partial integration when taking into account the correspondence between sums and integrals and between differences and derivatives. This motivates the name.

Abelian inequality

Is a monotonically decreasing sequence of positive sequence elements, applies ie

And the numbers are arbitrary real ( or complex ), then

(For the notation " max" see the largest and smallest element. )

This statement follows directly by applying the triangle inequality to the right side of the above equation for the abelian partial summation.

Example of use

Abel used the inequality in his work (see sources), to prove that a power series

For a given positive real number converges for each small positive number and is convergent to a continuous function. The key step here is the transformation

And there is a decreasing sequence, you can see the sum on the right side of the abelian inequality by

Estimate upward, and the two factors are arbitrarily small for large.

Swell

• H. Heuser, Textbook of Analysis, 9th edition, Stuttgart 1991. ISBN 3-519-22231-0
• Niels Henrik Abel, studies on the series