\section*{Task 1 - 321011}

Bernd is using a simple calculator. For the calculation 1:7, he gets the result 0.1428571, shown to 7 decimal places.

Now he thinks: The true decimal fraction can be found without performing any further numerical calculations. However, it is also possible to find it with just one simple additional calculation step.

Describe and justify how the desired decimal fraction can be found using one of these methods!
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In the well-known written method for division by 7, if the method does not end with a remainder of 0, at most six different remainders can occur. As soon as the same remainder appears twice, a repetition of the respective subsequent result digits occurs. Therefore, the resulting decimal fraction, if it does not terminate after at most 6 digits, must be periodic-infinite with a period length that is not greater than 6.

The calculator result 0.1428571 shows (regardless of whether the last digit 1 is due to rounding or truncation without rounding):

Starting with the tenths digit 1, six different digits are obtained, including the sixth non-rounded digit 7, and no termination follows. Therefore, the sought true decimal fraction can only be the periodic-infinite decimal fraction \(0, \overline{142857}\).