Abel's criterion

The Abel's test is a mathematical convergence criterion for an infinite series . It belongs to the group of direct criteria and was named after the Norwegian mathematician Niels Henrik Abel (1802–1829).

Abelian criterion for convergence

The series with converges when of finite variation and the series is convergent. ${\displaystyle \sum _{k=1}^{\infty }a_{k}b_{k}}$ ${\displaystyle a_{k},b_{k}\in \mathbb {R} }$ ${\displaystyle (a_{k})}$ ${\displaystyle \sum _{k=1}^{\infty }b_{k}}$

In the real world, the requirement that is monotonous is sufficient instead of the finite variation of . ${\displaystyle a_{k}}$ ${\displaystyle \lim _{k\to \infty }a_{k}\neq \pm \infty }$ ${\displaystyle a_{k}}$

Abelian criterion for uniform convergence

Be

${\displaystyle A=(a_{n}\colon D\to \mathbb {R} )_{n=1,2,\ldots }}$

and

${\displaystyle B=(b_{n}\colon D\to \mathbb {R} )_{n=1,2,\ldots }}$

sequences of functions defined in the field . be uniformly bounded , the sequences for each monotonous and the series ${\displaystyle D}$ ${\displaystyle A}$ ${\displaystyle (a_{n}(x))_{n=1,2,\ldots }}$ ${\displaystyle x\in D}$

${\displaystyle \sum _{n=1}^{\infty }b_{n}(x)}$

convergent uniformly , then it is also the series

${\displaystyle \sum _{n=1}^{\infty }a_{n}(x)b_{n}(x)}$

uniformly convergent.

Application in practice

In practice, one tries to factor the individual summands of an infinite series with the help of the Abel criterion in such a way that one of the factors results in a known convergent series and the others a monotonically decreasing sequence of positive numbers.

See also

• Dirichlet's criterion

Individual evidence

1. ↑ Fichtenholz G., differential and integral calculus , ISBN 978-3-8171-1279-1 , Volume 2, XII., §1.