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#1 2011-04-23 23:28:09
- UltraGnosis
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Another phi curiosity
I was playing around with the series n + 1/n + 1/n² + 1/n³... in a spreadsheet and noted that it converges to n + 1/(n-1) for n > 0. Putting n = phi obviously gives the convergence of 2 x phi. However, stopping the sum at the 5th term, I was surprised to see that phi + 1/phi + 1/phi^2 + 1/phi^3 + 1/phi^4 = 3 exactly (well, to 14 places at least). I'm wondering whether:
a) Anyone's ever noted this before
b) There's anything deeper here, such as a geometric explanation
#2 2011-04-24 05:46:33
- bobbym
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Re: Another phi curiosity
Hi UltraGnosis;
That is a very good piece of experimental math.
a) I did not know that before.
b) Sorry, my field is numerical analysis, so I am what they call a thrasher. Geometric ideas rarely penetrate my skull But there is an algebraic reason for your discovery.
On its way to becoming:
Your sum briefly stops at exactly 3. That is the only integer it ever touches.
If we substitute phi into your finite sum of
We get:
In mathematics, you don't understand things. You just get used to them.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#3 2011-04-24 16:29:35
- UltraGnosis
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Re: Another phi curiosity
Hi bobbym,
Thanks for the reply. The surprise for me is all those odd powers of an irrational conspiring to yield an integer! Putting the next phi-like term in the series, that is (3 + √5)/2, for n leaves the convergence unchanged at 2 x phi but now yields 3 when truncated to only 2 terms, leaving a somewhat more trivial algebra of (3 + √5)/2 + 2/(3 + √5) = 3.
#4 2011-04-24 19:01:47
- bobbym
- Administrator
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Re: Another phi curiosity
Hi;
Have you tried the next one?
In mathematics, you don't understand things. You just get used to them.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#6 2011-04-25 06:50:40
- bobbym
- Administrator
Online
Re: Another phi curiosity
Hi;
Now you can vary the parameter a very quickly. So far
is the only one that converges to an integer.
In mathematics, you don't understand things. You just get used to them.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.