Abelian partial summation

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

statement

Let it be a natural number and real numbers . Then applies ${\displaystyle n}$ ${\displaystyle a_{1},a_{2},\ldots ,a_{n},b_{1},b_{2},\ldots ,b_{n}}$

${\displaystyle \sum _{k=1}^{n}a_{k}b_{k}=A_{n}b_{n}+\sum _{k=1}^{n-1}A_{k}(b_{k}-b_{k+1})}$

With

${\displaystyle A_{k}=a_{1}+a_{2}+\ldots +a_{k}.}$

The statement has a certain formal similarity to partial integration , if one takes into account the correspondence between sums and integrals and between differences and derivatives. This motivates the name.

Abelian inequality

Is a monotonically decreasing sequence with positive followers, i. H. applies ${\displaystyle (b_{k})}$

${\displaystyle b_{1}\geq b_{2}\geq b_{3}\geq \ldots \geq b_{n}>0,}$

and if the numbers are arbitrarily real (or complex ), then applies ${\displaystyle a_{k}}$

${\displaystyle {\bigg |}\sum _{k=1}^{n}a_{k}b_{k}{\bigg |}\leq b_{1}\cdot \max _{k=1,\ldots ,n}|A_{k}|.}$

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

This statement follows directly by applying the triangle inequality to the right-hand side of the equation given above for the Abelian partial summation.

Application example

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

${\displaystyle a_{0}+a_{1}x+a_{2}x^{2}+\ldots ,}$

which converges for a certain positive real number , is also convergent for every smaller positive number and represents a continuous function. The essential step is the forming ${\displaystyle x=x_{0}}$ ${\displaystyle x<x_{0}}$ ${\displaystyle 0<x<x_{0}}$

${\displaystyle a_{m}x^{m}+a_{m+1}x^{m+1}+\ldots ={\Big (}{\frac {x}{x_{0}}}{\Big )}^{m}\cdot a_{m}x_{0}^{m}+{\Big (}{\frac {x}{x_{0}}}{\Big )}^{m+1}\cdot a_{m+1}x_{0}^{m+1}+\ldots ,}$

and since it is a monotonically decreasing sequence, one can calculate the sum on the right-hand side according to the Abelian inequality ${\displaystyle (x/x_{0})^{k}}$

${\displaystyle {\bigg |}{\frac {x}{x_{0}}}{\bigg |}^{m}\cdot \sup _{k\geq m}{\bigg |}\sum _{\nu =m}^{k}a_{\nu }x_{0}^{\nu }\,{\bigg |}}$

estimate upwards, and the two factors become arbitrarily small for large . ${\displaystyle m}$

swell

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

${\displaystyle \textstyle 1+{\frac {m}{1}}\cdot x+{\frac {m\,\cdot \,(m-1)}{1\,\cdot \,2}}\cdot \,x^{2}+{\frac {m\cdot \,(m-1)\,\cdot \,(m-2)}{1\,\cdot \,2\,\cdot \,3}}\cdot \,x^{3}+\ldots }$,

J. Reine Angew. Math. 1 (1826) 311-331

The Abelian inequality together with the relevant transformation can be found as Theorem III on p. 314.

Web links

Abelian inequality