concept notes on harmonic series and p-series
(This post is temporary.)
The infinite sum of 1/np (p-series) converges for all p greater than 1.
But why does the harmonic series (p=1) diverge?
"For sum(an) to converge it is necessary but not sufficient that an converge to zero as n goes to infinity."
The first proof on this page helps:
http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29
The graph is very subtle, like a logarithm.
Sigh.
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + ... + 1/999999999 + ...
I think there is no conceptual explanation.
Definition of convergence of a series:
http://www.shu.edu/projects/reals/numser/defs/convser.html
So perhaps the harmonic series does not converge because of the definition of a limit.
Will investigate.
The infinite sum of 1/np (p-series) converges for all p greater than 1.
But why does the harmonic series (p=1) diverge?
"For sum(an) to converge it is necessary but not sufficient that an converge to zero as n goes to infinity."
The first proof on this page helps:
http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29
The graph is very subtle, like a logarithm.
Sigh.
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + ... + 1/999999999 + ...
I think there is no conceptual explanation.
Definition of convergence of a series:
http://www.shu.edu/projects/reals/numser/defs/convser.html
So perhaps the harmonic series does not converge because of the definition of a limit.
Will investigate.