6.19 Solve the equation

$1+2 x+4 x^{2}+\ldots+(2 x)^{n}+\ldots=3.4-1.2 x$, given that $|x|<0.5$.
6.19 The left side of the equation is the sum of an infinite geometric progression, where \( b_{1}=1 \) and \( |q|=|2 x|<1 \), since \( |x|<0.5 \). According to formula (6.10), we have

\[
\begin{aligned}
& 1+2 x+4 x^{2}+\ldots=\frac{1}{1-2 x} \\
& \frac{1}{1-2 x}=3.4-1.2 x ;(3.4-1.2 x)(1-2 x)=1
\end{aligned}
\]

\[
2.4 x^{2}-8 x+2.4=0 ; 3 x^{2}-10 x+3=0
\]

from which \( x_{1}=3, x_{2}=\frac{1}{3} \). Only the root \( x=\frac{1}{3} \) satisfies the condition.

Answer: \( x=\frac{1}{3} \).