In this series, I discuss some ways of convincing students that and that, more generally, a real number may have more than one decimal representation even though a decimal representation corresponds to only one real number. This can be a major conceptual barrier for even bright students to overcome. I have met a few math majors within a semester of graduating — that is, they weren’t dummies — who could recite all of these ways and were perhaps *logically* convinced but remained *psychologically *unconvinced.

**Method #4**. This is a direct method using the formula for an infinite geometric series… and hence will only be convincing to students if they’re comfortable with using this formula. By definition,

This is an infinite geometric series. Its first term is , and the common ratio needed to go from one term to the next term is . Therefore,

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*Posted by John Quintanilla on September 3, 2013*

https://meangreenmath.com/2013/09/03/why-does-0-999-1-part-3/

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