Today I have played around with infinite series as well as with the famous Zeta function. And I did a litte proof that the zeta-function converges for and . I am new in this area of mathmatics and therefore I want to check (by contributing it) wether this kind of argumentation is valid.
The Zeta function is definded by:
Let be the -th term of :
This convergence was proven by Euler!
Let be the -th term of , where is a real number greater than :
For now we don't know wether this converges or not!
We know that for all natural numbers . Therefore is bounded below at and also non decreasing. For each natural number and each real number greater than it is obviously true, that:
Therefore is also bounded above at and because is bounded above, bounded below and is non-decrasing, it must converge.
converges for all real numbers
Now we can start with complex values for ...
Let be a complex value for the Zeta-function and let be the -th term of the Zeta-function, where :
Now there's a little bit algebra:
Because is a natural number we can simplify this a tiny bit:
There is a convergency test for complex infinite series, which states:
If converges absolutely and for every natural number then converges as well.
Because , converges absolutely.
This inequality for real numbers was proven in the first section of this note. Therefore must converge. And finally:
converges for all complex numbers where .
I know that there is a little gap in the domain of the Zeta-function for which the convergence isn't proven by this note (all complex number where ), but this note is just a result of a boring afternoon. I'm wondered wether you can prove this gap with the same kind of argumentation or wether you will need a new base for the complete proof. (I even don't know wether this proof is valid, that's the main reason why I contribute this note)