The discussion centers around the proof of the convergence of the series ##\sum_{m=2}^\infty \sum_{n=2}^\infty 1/n^m## and whether it equals 1. Participants explore various approaches and calculations related to this series, including numerical simulations and algebraic manipulations.
Participants express differing views on the nature of the series and its convergence, with some supporting the idea that it sums to 1 while others raise questions about the underlying assumptions and interpretations.
There are indications of missing assumptions regarding convergence criteria and the definitions of the series involved. The discussion does not resolve these uncertainties.
Readers interested in series convergence, mathematical proofs, and numerical analysis may find this discussion relevant.
The series ##\sum_{n=2}^\infty (n(n-1))^{-1} = \sum_{n=2}^\infty \left( \frac{1}{n-1} - \frac1n \right)## does sum to 1 because ##\sum_{n=2}^N \left( \frac{1}{n-1} - \frac1n \right) =1-1/N##BWV said:OK, playing with Wolfram Alpha, the series reduces to ##\sum_{n=2}^\infty (n(n-1))^{-1}##, which does sum to 1