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Question

Use geometric series to convert  

 to a rational number:

1. When expressed as a geometric series, the first term is: 
2. When expressed as a geometric series, the common ratio is: 
3. The infinite geometric sum, when expressed as a rational number is: 

Answer 1

0.99999999... = 0.9 + 0.09 + 0.009 + 0.0009 + ...

This is a geometric series with first term 0.9 and common ratio 0.1

Since the common ratio is between -1 and 1, the series has a finite sum.

Sum = (first term) / [1 - common ratio] = 0.9 / [1 - 0.1} = 0.9 / 0.9 = 1