sequences and series - The sum $1+frac{1}{3}+frac{1}{5}+frac{1}{7}+cdots-(frac{1}{2}+frac{1}{4}+frac{1}{6}+cdots)$ does not exist.

What are the argument(s) that I can use proving that

$$1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots-(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\cdots)$$

does not exist.

The question was:

Find a arrangement of $\sum\frac{(-1)^{n-1}}{n}$ for which the new sum is not exist(even not $+\infty$ or $-\infty$)