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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.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

## #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

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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.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

## #5 2011-04-25 01:04:08

UltraGnosis
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### Re: Another phi curiosity

Ah yes! It actually converges to 4.

## #6 2011-04-25 06:50:40

bobbym

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### 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.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.