Prove that $ \lim_{N\to \infty} \sum_{n=1}^N \frac{1}{\phi(n)} $ exists or does not exist Prove that the limit exists or does not exist:
$$ \lim_{N\to \infty} \sum_{n=1}^N \frac{1}{\phi(n)}, $$
where $\phi(n)$ is the Euler Totient function.
The ratio test was inconclusive.
I'm fairly sure the p-series test says this series diverges because $p=1$ but then again in this case I'm not sure how to deal with a function in the place where $n$ normally is.

Since $\varphi(n)\leq n$ it follows that $$\frac{1}{\varphi(n)}\geq \frac{1}{n}$$ and hence $\sum \frac{1}{\varphi(n)}=\infty$.
