Question

0,1+0,01+0,001+...+0 00...(100 de 0)01. Ajutor ofer coroana si puncte

Answer 1

[tex]0,1+0,01+0,001+....+0,0.....01=\\ \frac{1}{10}+\frac{1}{100}+\frac{1}{100}+...+\frac{1}{10^{101}}=\\ \frac{10^{100}+10^{99}+10^{98}+.....+1}{10^{101}}=\\ Luam\ suma\ separat:\\ S=1+10+10^2+....+10^{100}|\cdot(-10)\\ -10S=-10-10^2-......-10^{101}\\ ....................................................+\\ -9S=1-10^{101}\\ S=\frac{10^{101}-1}{9}\\ Revenim:\frac{\frac{10^{101}-1}{9}}{10^{101}}=\frac{10^{101}-1}{9\cdot 10^{101}}[/tex]