# Converting an Infinite Decimal Expansion to a Rational Number

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## Question

Express



as a ratio of integers.



## Explanation of Each Step

### Step (1)

Although not necessary, writing the repeating decimal expansion into a few terms of an infinite sum allows us to see more clearly what we need to do: relate each term to each other in some way to write the given number using sigma notation.

### Step (2)

Each term in the sum is equal to 79 times 10 to a power. Explicitly writing out what these powers are helps us look for a pattern in the individual terms of our sum.

### Step (3)

Suppose we allow our infinite series to start with the term . Then each term in the infinite series (after the second term) is related to its previous term by a factor of . Using the form of the geometric series:



and substituting our identified values for  into this formula yields Step (3).

### Step (4)

Here, we apply our formula for the sum of an infinite series,



The rest of the problem is algebraic manipulation of fractions to find a simplified ratio of two integers.

## Possible Areas of Confusion

### Getting Started

Any problem of this type could be started in the same way: by writing out the first few terms of an infinite series. This way, it is easier to see a pattern in the terms of the infinite series.

Essentially, we solved the given problem by writing  as , which isolated the repeating digits, which can be written as a geometric series.

### Step (3)

Recall that our general form of the geometric series is



In Step (2), we can identify , and . We may substitute these values into the above general form to obtain Step (3).

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Next: Finding the Sum of an Infinite Series