I am taking honors Calculus II and have been doing reasonably well in the course until the current problem set which is due tomorrow. One exercise that is really giving me trouble is this:
Prove that $\sum\limits_{k=1}^\infty \frac{1}{k^5}$ is irrational.
It's easy to prove that the series converges by the p-test, but I I just don't see where to go with proving its irrationality. Any hints would be greatly appreciated, but please don't give the whole problem away.
I don't really know why my professor likes such problems, but we spent the last two lectures proving that things were irrational. We proved first that numbers like $\sqrt{2}$, $\log_2{3}$, and $\sqrt{2}+\sqrt{3}$ are irrational. Then last lecture we proved that $\pi$ is irrational.
But for the problem above, I tried to prove it by contradiction,
Suppose that $\sum\limits_{k=1}^\infty \frac{1}{k^5}$ is rational, then there exist $a,b$ such that $a \in \mathbb{Z}$ and $b \in \mathbb{Z}\setminus\{0\}$ such that $$\sum\limits_{k=1}^\infty \frac{1}{k^5} = \frac{a}{b}.$$ I really don't know where to go from here. I looked over my notes and we haven't done anything like this. The only thing that we did that was kind of similar was proving that $\sum\limits_{k=1}^n \frac{1}{k}$ is never an integer for $n > 1$, but this is only a finite series and it's dealing with an integer. Maybe there are related, but I don't know how I could use unique factorization to prove the above irrationality of the given series. Would that work?
This is an unsolved problem! See the Wikipedia article on Apéry's theorem. Zudilin showed in 2001 that at least one of the numbers $\zeta(5),\zeta(7),\zeta(9)$, and $\zeta(11)$ is irrational, and this appears to be the best result so far. (Here $\zeta(n)=\sum_{k=1}^\infty \frac{1}{k^n}$.)