Just because you moved the parenthesis around doesn't make the result different.
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Anonymous2007-01-26 17:27
>>6
Truth, as per the associative property of addition. Thus, 1 = 0.
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Anonymous2007-01-26 17:49
>>7
Associative property of addition says a + (b + c) = (a + b) + c. This can be extended by induction in various ways to any finite sum, but not necessarily to infinite sums.
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Anonymous2007-01-26 20:55
>>8
For the simple reason that the OP provided for us.
>>4
1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ... = 1. What we are learning ITT is that
1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ... != (1 + -1) + (1 + -1) + (1 + -1) + (1 + -1) + ...
.
By the way, there are only three dots in an ellipsis.
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Anonymous2007-01-26 21:08
If you can come up with a field where the additive identity element and the multiplicative identity element are equal, then yes.. 1 = 0 IN THAT FIELD.
Note this field isn't the real numbers (is that even a field?)
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Anonymous2007-01-26 21:39
>>10
Er, wouldn't you only have that [the additive identity element in that field] = [the multiplicative identity element in that field] in that field? Even if they were called 1 and 0 respectively, they wouldn't be the same numbers we know.
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Anonymous2007-01-26 21:40
>>10
It is by definition not a field. 0 and 1 are required to be different elements as part of the field axioms. Furthermore, even if you ignore that axiom, you end up with the trivial field {0} being the only one in which the additive and multiplicative identities are equal.
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Anonymous2007-01-26 22:29
>>4
Not necessarily. The Cesaro summation of that series is 1/2.