1. The solution to the equation $\frac{1+3^{x}+3^{2 x}+\cdots+3^{2010 x}}{1+3^{-x}+3^{-2 x}+\cdots+3^{-2010 x}}=3$ is $x=$
$-1 \cdot \frac{1}{2010}$.
The original equation is transformed into
$$
\begin{array}{l}
\frac{3^{2010 x}\left(3^{-2010 x}+3^{-2000 x}+\cdots+3^{-x}+1\right)}{1+3^{-x}+3^{-2 x}+\cdots+3^{-2010 x}}=3 \\
\Rightarrow 3^{2010 x}=3 \Rightarrow x=\frac{1}{2010} .
\end{array}
$$