# (a) write the repeating decimal as a geometric series, and (b) write its sum as the ratio of two integers. 0.overline{81}

Question
Series
(a) write the repeating decimal as a geometric series, and (b) write its sum as the ratio of two integers. $$0.\overline{81}$$

2021-01-01
Given decimal: $$0.\overline{81}$$
(a) For the repeating decimal 0.81, we can write
$$0.\overline{81}=0.818181...$$
$$=0.81+0.0081+0.000081+0.00000081+...$$
$$=\frac{81}{10^2}+\frac{81}{10^4}+\frac{81}{10^6}+...$$
$$=\sum_{n=0}^{\infty}\frac{81}{10^2}(\frac{1}{10^2})^n$$ (b) Given Series: $$=\sum_{n=0}^{\infty}\frac{81}{10^2}(\frac{1}{10^2})^n$$
Given series is a Geometric series with ratio $$r=\frac{1}{10^2}$$ and $$a_1=\frac{81}{10^2}$$
So the sum is $$=\frac{a_1}{(1-r)}$$
$$=\frac{\frac{81}{100}}{(1-\frac{1}{100})}$$
$$=\frac{81}{99}$$
$$=\frac{9}{11}$$
Result
$$\frac{9}{11}$$

### Relevant Questions

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