Subseries of the Harmonic Series.
Catalin Ciuperca
Department of Mathematics
5pm, November 14, 2008
Minard 334
Abstract: It is well known that the series
1+1/2+1/3+...+1/n+...
is divergent, i.e. the sums (1+1/2+...+1/n) increase without bound. What
happens if we strike out those terms whose denominators contain the
digit 9 and consider the series
1+1/2+1/3+...+1/8+1/10+1/11+...+1/18+1/20+1/21+... ?
Surprisingly, this series converges. What about the series
1/2+1/3+1/5+1/7+1/11+...
where the denominators are all the prime numbers? This is actually
divergent.
In this talk we discuss these examples as well as other subseries of
the
harmonic series.