What is wrong with the sum of these two series?
Solution 1:
The first series only applies when $|x| < 1$ whereas the second series only applies when $\left|\frac{1}{x}\right| < 1$ (i.e. $|x| > 1$). By adding them, you are assuming that they both apply simultaneously, but they don't (for any $x$).
Solution 2:
The domains of convergence of these two sequences don't coincide. One converges for $|x|>1$ and the other for $|x| < 1$. Therefore, the sum is meaningless.