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Unread 29-08-2010, 19:05
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Ether Ether is offline
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Re: 0.9 repeating = 1?

Quote:
Originally Posted by ExTexan View Post
I'm not sure anything was proven! :confused
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:

Quote:
A real number L is said to be the limit of the sequence Xn if and only if for every real number ε > 0, there exists a natural number N such that for every n > N we have | Xn−L | < ε.
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.




Last edited by Ether : 29-08-2010 at 19:29.
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