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

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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 106,353

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.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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

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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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 106,353

Hi;

Have you tried the next one?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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

Ah yes! It actually converges to 4.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 106,353

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.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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