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.
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.
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).
Förkunskapskrav Basic courses in algebra and calculus.
Examinator Maria Roginskaya, Marina Axelson-Fis