Express the number as a ratio of integers.
To express: The given number as a ratio of integers.
The number 0.8¯ can be expressed as 89.
The geometric series ∑n=1∞arn−1 (or) a+ar+ar2+⋯ is convergent if |r|<1 and its sum is a1−r, where a is the first term and r is the common ratio of the series.
Rewrite the number and express 0.8¯ as follows,
Clearly, it is geometric series with first term of the series is a=810 and common ratio is
Since |r|<1 and the Result stated above, the geometric series ∑n=1∞810n is convergent.
Obtain the sum of the geometric series.
Since a=810 and r=110,
Thus, the series is convergent to the sum is ∑n=1∞810n=89.
Therefore, the number 0.8¯ can be expressed as 89.