How can I prove the following using power series? :
Prove that the number whose decimal expression is given by ''$e_n...e_1,\hspace{1mm}d_1...d_m\hspace{1mm}\overline{p_1...p_s}$'' where $e_j,d_i,p_k$ (for every $j=1,...,n$; $i=1,...,m$; $k=1,...,s$) represents a digit from 0 to 9, and $\overline{p_1...p_s}$ is the period, it is rational.
I'm really having trouble with this.
Hint:
$e_n...e_1.d_1...d_m\overline{p_1...p_s} = e_n...e_1.d_1...d_m + 0.\underbrace{0...0}_\text{m times}\overline{p_1...p_s} $ .
Then representing the second factor as $x$, subtract $10^{m+s} x $ and $x$ , What do you obtain?
To solve the problem you can finish and adapt to power series this argument.