Question

Consider the following repeating decimal. 0.619

Answer 1

Answer:

a.  0.6$\overline {19}$ = $0.6 + \ \sum \limits ^{\infty}_{n=0} \ 0.019 \ ( \dfrac{1}{100})^n$

b. 0.6$\overline {19}$  = $\mathbf{\dfrac{613}{990}}$  

Step-by-step explanation:

Consider the following repeating decimal. 0.6$\overline {19}$

a) Write the repeating decimal as a geometric series.

0.6$\overline {19}$ is being expressed as 0.6191919...

0.6$\overline {19}$ = 0.6 + 0.019 + 0.00019+ 0.0000019 + ...

0.6$\overline {19}$ =$0.6 + \dfrac{19}{1000}+ \dfrac{19}{100000}+ \dfrac{19}{10000000}+ ...$

0.6$\overline {19}$ = $0.6+ \dfrac{19}{1000} \begin {bmatrix} 1 + \dfrac{1}{100} + \dfrac{1}{10000}+ ... \end {bmatrix}$

0.6$\overline {19}$ = $0.6 + \ \sum \limits ^{\infty}_{n=0} \ 0.019 \ ( \dfrac{1}{100})^n$

(b) Write the sum of the series as the ratio of two integers

0.6$\overline {19}$  = $0.6 + 0.019 ( \dfrac{1}{1-0.01})$

0.6$\overline {19}$  =$0.6 + \dfrac{19}{1000}\times \dfrac{100}{99}$

0.6$\overline {19}$  = $0.6 + \dfrac{19}{990}$

0.6$\overline {19}$  = $\dfrac{594+19}{990}$

0.6$\overline {19}$  = $\mathbf{\dfrac{613}{990}}$