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#1 2011-04-23 01:28:09

UltraGnosis
Member
Registered: 2011-04-23
Posts: 3

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

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#2 2011-04-23 07:46:33

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,771

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#3 2011-04-23 18:29:35

UltraGnosis
Member
Registered: 2011-04-23
Posts: 3

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.

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#4 2011-04-23 21:01:47

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,771

Re: Another phi curiosity

Hi;

Have you tried the next one?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#5 2011-04-24 03:04:08

UltraGnosis
Member
Registered: 2011-04-23
Posts: 3

Re: Another phi curiosity

Ah yes! It actually converges to 4.

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#6 2011-04-24 08:50:40

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,771

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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