power series. 0.999... = sigma 9/10^n n=1 -> inf = 9 * sigma 1/10^n n=1 -> inf = 9 * ((sigma 1/10^n n=0 -> inf) - 1) = power series sigma 1/10^n n=0 -> inf =

power series. 0.999... = sigma 9/10^n n=1 -> inf = 9 * sigma 1/10^n n=1 -> inf = 9 * ((sigma 1/10^n n=0 -> inf) - 1) = power series sigma 1/10^n n=0 -> inf = 1/(1-1/10) = 9 * (1/(1-1/10) - 1) = 1 qed.