Answer:
a. 0.6[tex]\overline {19}[/tex] = [tex]0.6 + \ \sum \limits ^{\infty}_{n=0} \ 0.019 \ ( \dfrac{1}{100})^n[/tex]
b. 0.6[tex]\overline {19}[/tex] = [tex]\mathbf{\dfrac{613}{990}}[/tex]
Step-by-step explanation:
Consider the following repeating decimal. 0.6[tex]\overline {19}[/tex]
a) Write the repeating decimal as a geometric series.
0.6[tex]\overline {19}[/tex] is being expressed as 0.6191919...
0.6[tex]\overline {19}[/tex] = 0.6 + 0.019 + 0.00019+ 0.0000019 + ...
0.6[tex]\overline {19}[/tex] =[tex]0.6 + \dfrac{19}{1000}+ \dfrac{19}{100000}+ \dfrac{19}{10000000}+ ...[/tex]
0.6[tex]\overline {19}[/tex] = [tex]0.6+ \dfrac{19}{1000} \begin {bmatrix} 1 + \dfrac{1}{100} + \dfrac{1}{10000}+ ... \end {bmatrix}[/tex]
0.6[tex]\overline {19}[/tex] = [tex]0.6 + \ \sum \limits ^{\infty}_{n=0} \ 0.019 \ ( \dfrac{1}{100})^n[/tex]
(b) Write the sum of the series as the ratio of two integers
0.6[tex]\overline {19}[/tex] = [tex]0.6 + 0.019 ( \dfrac{1}{1-0.01})[/tex]
0.6[tex]\overline {19}[/tex] =[tex]0.6 + \dfrac{19}{1000}\times \dfrac{100}{99}[/tex]
0.6[tex]\overline {19}[/tex] = [tex]0.6 + \dfrac{19}{990}[/tex]
0.6[tex]\overline {19}[/tex] = [tex]\dfrac{594+19}{990}[/tex]
0.6[tex]\overline {19}[/tex] = [tex]\mathbf{\dfrac{613}{990}}[/tex]
