<s>[INST] What is wrong with the sum of these two series? Could anyone help me to find the mistake in the following problem? Based on the formula of the sum of a geometric series:
\begin{equation}
1 + x + x^{2} + \cdots + x^{n} + \cdots = \frac{1}{1 - x}
\end{equation}
\begin{equation}
1 + \frac{1}{x} + \frac{1}{x^{2}} + \cdots + \frac{1}{x^{n}} + \cdots = \frac{1}{1 - 1/x} = \frac{x}{x-1}
\end{equation}
Adding both equations
\begin{equation}
2 + x + \frac{1}{x} + x^{2} + \frac{1}{x^{2}} + \cdots + x^{n} + \frac{1}{x^{n}} + \cdots = \frac{1}{1 - x} + \frac{x}{x-1} = \frac{1-x}{1-x} = 1
\end{equation}
So,
\begin{equation}
2 + x + \frac{1}{x} + x^{2} + \frac{1}{x^{2}} + \cdots + x^{n} + \frac{1}{x^{n}} + \cdots = 1
\end{equation}
And the left side is always bigger than $2$ for $x>0$. 
What is wrong?? Thanks in advance
 [/INST] 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.
</s>