Like the series 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... = ln2.
ln2 is a real number. Approximately equal to .6931
if you re-arrange the series, say like:
1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 ...
you still use up all of the same terms. However, the sum will be greater.
The real weird thing about this. One can simply pick a number. Any number.. and one can find a re-arrangement of the series which will sum to that number.
It might be a *tad* difficult to list the series correctly, but it does exist.
If you want the number to be infinity in the extended real number system.. simply choose to sum all of the positive terms first.. 1 + 1/3 + 1/5 + 1/7 + 1/9 + ...
This series diverges. In other words, is not bound by any real number.
It "blows up" even before one gets a chance to add on any negative terms..
So far, except for infinity, the extended real numbers and such, I haven't used anything here past high school algebra. We're just adding fractions.
of course to show the sum of 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... is really ln2, requires a tad more. but one could take a calculator, start adding and subtracting fractions and observe the sum gets closer and closer to 0.6931 as one keeps it up..
Did you know, there are two decimal representations of the number 1?
maybe or maybe not. They might look different, but they are in fact the same thing..
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waysider
Say what?
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Ham
Like the series 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... = ln2.
ln2 is a real number. Approximately equal to .6931
if you re-arrange the series, say like:
1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 ...
you still use up all of the same terms. However, the sum will be greater.
The real weird thing about this. One can simply pick a number. Any number.. and one can find a re-arrangement of the series which will sum to that number.
It might be a *tad* difficult to list the series correctly, but it does exist.
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Ham
If you want the number to be infinity in the extended real number system.. simply choose to sum all of the positive terms first.. 1 + 1/3 + 1/5 + 1/7 + 1/9 + ...
This series diverges. In other words, is not bound by any real number.
It "blows up" even before one gets a chance to add on any negative terms..
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Ham
I flunked Real Analysis my first time through..
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Ham
I always liked series and sequences.
So far, except for infinity, the extended real numbers and such, I haven't used anything here past high school algebra. We're just adding fractions.
of course to show the sum of 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... is really ln2, requires a tad more. but one could take a calculator, start adding and subtracting fractions and observe the sum gets closer and closer to 0.6931 as one keeps it up..
Did you know, there are two decimal representations of the number 1?
maybe or maybe not. They might look different, but they are in fact the same thing..
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waysider
It's out of my league, Ham.
Visual patterns-yes.
Audio patterns-yes.
Numeric sequencing patterns-nope.
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krys
I sorta get the gist of what you're saying....but I will have to take your word for it.
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TrustAndObey
I think my brain has been re-arranged sufficiently by this information that I no longer have a clue of what is being said!
Thank you, may I have another!
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