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Re: 0.9 repeating = 1?
Quote:
Originally Posted by Ether
The mathematical meaning of the repeating decimal .999... is a limit.
It is the limit of the sequence of partial sums of the infinite series 9/10 + 9/100 + 9/1000 + ...
The sequence of partial sums of the above series is equal to (1-1/10), (1-1/100), (1-1/1000), ... (1-1/10^n)
Using the definition of limit of a sequence:
It can be shown that the limit of the above sequence is "1".
Therefore, "1" and ".999..." mean exactly the same thing. They are two different ways of writing the same real number.
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The limit of a series of partial sums is not equal to the actual sum at all. This is by definition of a limit. A limit is what the function must approach ever closer without ever reaching it. If it ever actual reaches it at any point, then it is not truly its limit.
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