real analysis - Prove that $lim_{Nto infty} sum_{n=1}^N frac{1}{phi(n)}$ 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.

Answer

Since $\varphi(n)\leq n$ it follows that

$$\frac{1}{\varphi(n)}\geq \frac{1}{n}$$ and hence $\sum \frac{1}{\varphi(n)}=\infty$.