#### To determine

**To express:** The given number as a ratio of integers.

#### Answer

The number 1.234567¯ can be written as 45,67937,000.

#### Explanation

**Given:**

1.234567¯=1.234567567567...

**Result used:**

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.

**Calculation:**

Rewrite the number and express 1.234567¯ as follows,

1.234567¯=1.234+0.000567¯=1.234+(0.000567+0.000000567+⋯)=1.234+(567106+567109+⋯)

1.234567¯=1.234+∑n=2∞567103n (1)

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,

∑n=2∞567103n=5671061−1103=567106999103=567106×103999=2137,000

Thus, the series is convergent to the sum ∑n=2∞567103n=2137,000. (2)

Substitute equation (2) in equation (1),

1.234567¯=1.234+2137,000=1.234(37,000)+2137,000=45,658+2137,000=45,67937,000

Therefore, the number 1.234567¯ can be written as 45,67937,000.