5 added 52 characters in body

Also I can't delete messages - I meant $(-1)^{n+1}$.

As a matter of fact your sum diverges, a little manipulation shows that $$\sum_{n \leq X} \frac{(-1)^n}{\phi(n)} = \sum_{n \leq X, 2|n} \frac{1}{\phi(n)} - \sum_{n \leq X, (n,2)=1} \frac{1}{\phi(n)} = \sum_{n \leq X/2} \frac{1}{\phi(2n)} -\sum_{n \leq X,(n,2)=1}\frac{1}{\phi(n)}$$ The above equala to $$\sum_{n \leq X/2, (n,2)=1} \frac{1}{\phi(2n)} + \sum_{n \leq X/2, 2|n} \frac{1}{\phi(2n)} - \sum_{n \leq X, (n,2)=1} \frac{1}{\phi(n)}$$ By multiplicativity of $\phi(n)$ we have $\phi(2n) = \phi(n)$ when $(n,2)=1$. Thus the first sum above is a sum over $1/\phi(n)$ and the above equation simplifies to $$- \sum_{X/2 < n \leq X, (n,2)=1} \frac{1}{\phi(n)} + \sum_{n \leq X/4} \frac{1}{\phi(4n)}$$ It follows that $$\sum_{n \leq X} \frac{(-1)^n}{\phi(n)}= -\sum_{X/2 < n \leq X, (n,2)=1} \frac{1}{\phi(n)} + \sum_{n \leq X/4} \frac{1}{\phi(4n)} \sim c \cdot \log{X}$$ because the first sum on the right converges to a constant, while the second sum on the right is asymptotically $c \cdot \log{X}$.

EDIT: Put details. , erased mention of an earlier confusion about $(-1)^{n+1}$ not being a multiplicative function :P (it is!)

4 added 651 characters in body

Also I can't delete messages - I meant $(-1)^{n+1}$. As a matter of fact your sum diverges, a little manipulation shows that $$\sum_{n \leq X} \frac{(-1)^n}{\phi(n)} = \sum_{n \leq X, 2|n} \frac{1}{\phi(n)} - \sum_{X/2 \sum_{n \leq X, (n,2)=1} \frac{1}{\phi(n)} = \sum_{n \leq X/2} \frac{1}{\phi(2n)} -\sum_{n \leq X,(n,2)=1}\frac{1}{\phi(n)}$$ The above equala to $$\sum_{n \leq X/2, (n,2)=1} \frac{1}{\phi(2n)} + \sum_{n \leq X/2, 2|n} \frac{1}{\phi(2n)} - \sum_{n \leq X, (n,2)=1} \frac{1}{\phi(n)}$$ By multiplicativity of $\phi(n)$ we have $\phi(2n) = \phi(n)$ when $(n,2)=1$. Thus the first sum above is a sum over $1/\phi(n)$ and the above equation simplifies to $$- \sum_{X/2 < n \leq X, (n,2)=1} \frac{1}{\phi(n)} + \sum_{n \leq X/4} \frac{1}{\phi(4n)}$$ The second It follows that $$\sum_{n \leq X} \frac{(-1)^n}{\phi(n)}= -\sum_{X/2 < n \leq X, (n,2)=1} \frac{1}{\phi(n)} + \sum_{n \leq X/4} \frac{1}{\phi(4n)} \sim c \cdot \log{X}$$ because the first sum on the right converges to a constant, while the third second sum on the right is asymptotically $c \cdot \log{X} \cdot (1+o(1))$, so the first sum diverges! log{X}$. EDIT: I just couldn't type this right. So many typo'sPut details. 3 added 59 characters in body Also I can't delete messages - I meant$(-1)^{n+1}$. As a matter of fact your sum diverges, a little manipulation shows that $$\sum_{n \leq X} \frac{(-1)^n}{\phi(n)} = -\sum_{X/2 < n \leq X, (n,2)=1} \frac{1}{\phi(n)} + \sum_{n \leq X/4} \frac{1}{\phi(n)}$$ frac{1}{\phi(4n)}$$The second sum converges to a constant, while the third is$c \cdot \log{X} \cdot (1+o(1))\$, so the first sum diverges!

EDIT: I just couldn't type this right. So many typo's.

2 added 320 characters in body; deleted 3 characters in body; added 9 characters in body
1