# MVEX01-18-18 Divergent series and summation methods

An infinite series is said to be convergent if the limit of its partial sums exists. However, it is easy to find an example of a divergent series. For instance, 1+2+3+…+n+… or 1-1+1-1+…+1-1+… Such divergent series puzzled mathematicians for a long time. The basic question was “Is there a reasonable way to define a ‘sum’ of a divergent series?” A lot of work in this direction was done by L. Euler (1707-1783), who laid the foundations for dealing with divergent series. However, not everyone considered these infinite sums as legitimate objects in mathematics. For example, in 1828 N. Abel wrote “Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever.” Despite of such statements divergent series gradually proved to be a useful tool in different areas of mathematics and physics.

The first explicit formulation of the theory of divergent series appeared in 1890 in the work of E. Cesaro. He suggested to replace the usual limit with the limit of average values, that is an infinite series is said to be (what it is now called) Cesaro convergent if the limit of the partial sum averages exists. Thus, it is easy to see that the series 1-1+1-1+…+1-1+… from above is Cesaro convergent to 1/2. However, the series 1+2+3+…+n+… is not Cesaro convergent. Cesaro’s idea was the very first example of a summation method, that is a rigorous way to assign a value to a divergent series. In the 20th century, the theory of divergent sequences grew significantly both at a technical level and through its applications.

The project consists of studying various summation methods (such as Cesaro, Abel, Riesz, Nörlund ones) as well as their interrelations and applications to Harmonic Analysis.​

Obs! För GU-studenter räknas projektet som ett projekt i Matematik (MMG900/MMG910).​​

Projektkod MVEX01-18-18
Gruppstorlek 3-4
Förkunskapskrav Basic courses in algebra and calculus.
Handledare Alexandr Usachev, usachev@chalmers.se.
Examinator Maria Roginskaya, Marina Axelson-Fis