Booth Id:
MATH012
Category:
Mathematics
Year:
2024
Finalist Names:
Rueter, Emma (School: Leibniz-Gymnasium Berlin)
Abstract:
In my research project I want to generalise the well-known concept of Riemann integration of functions on sequences in order to obtain a new tool for the investigation of sequences.
To do this, I define an integral for sequences based on the Riemann integral and then investigate which sequences are integrable. I prove necessary and sufficient conditions for sequence integrability. It turns out that integrability for a sequence is a stronger condition than cesàro-convergence: Every integrable sequence is cesàro-convergent, but not every cesàro-convergent sequence is also integrable. As a sufficient criterion for integrability, we find a condition that ressembles to Lebesgue integration: If suitable preimages under the sequence are "measurable", meaning that we can assign them a kind of length, the sequence is integrable. For sequences with finitely many accumulation points, we prove that this criterion is also necessary.