Express the number as a ratio of integers.
To express: The given number as a ratio of integers.
The number 1.234567¯ can be written as 45,67937,000.
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 1.234567¯ as follows,
Here, ∑n=2∞567103n is geometric series with first term of the series is a=567106 and common ratio is r=1103.
Since |r|<1 and the Result stated above, the geometric series ∑n=2∞567103n is convergent.
Obtain the sum of the geometric series.
Since a=567106 and r=1103, the series becomes,
Thus, the series is convergent to the sum ∑n=2∞567103n=2137,000. (2)
Substitute equation (2) in equation (1),
Therefore, the number 1.234567¯ can be written as 45,67937,000.