P-Series and Zeta #shorts

We look at examples of p-series. The main result is ithat the condition p larger than 1implies convergence. We also look at the zeta function zeta(s) which encodes all the p-series. The zeta function of course is not part of single variable calculus curricula but it is a fascinating topic still and worth mentioning because it relats to the by far most important open problem in mathematics: the Riemann hypothesis which makes a statement about the roots of the zeta function. In single variable calculus jargon: it is not known for which p values the series gives a zero value. The problem is that direct calculus tools only allow us to "see" the values where the series converges. For values like p=-1, where the series is 1 + 2 + 3 + 4 + .... the value -1/12 is only seen indirectly. The values for p smaller than one are behind a wall and like in other parts of science, we have to deduce knowledge about it indirectly using mathematical tools. The majority of mathematicians believe the Riemann hypothesis holds.