Alexa — Live Sequence Exploration

Date: 2026-03-29
Source: Alexa Amundson (direct)
Context: Raw exploration of G(n) boundary behavior and self-referential forms

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The Exploration

Starting from G(n) = n/(1+1/n)^n:

Boundary at n=0

n^n at n=0: 0^0 = 1 (convention)

1/0^0 = 1/1 = 1

G(0) = 0 (removable singularity, consistent)

Self-Referential Form

n/(n^n + n^n/n)^n — feeding G back into itself

The Sequence of n^(n+1)/(n+1)^n evaluated:

0^0 = 1 (boundary)

0^1/(1^0) = 0/1 = 0 → G(0) = 0 ✓

1^2/(2^1) = 1/2 → G(1) = 1/2 ✓

2^3/(3^2) = 8/9 → G(2) = 8/9 ✓

3^4/(4^3) = 81/64 → G(3) = 81/64 ✓

4^5/(5^4) = 1024/625 → G(4) = 1024/625 ✓

Pattern: n^(n+1) grows faster than (n+1)^n

The sequence crosses 1 between n=2 and n=3 (8/9 < 1, 81/64 > 1).
After that, G(n) grows linearly as n/e + 1/(2e).

Significance

The 0^0 = 1 convention is the algebraic anchor:

It makes G well-defined at the boundary

It connects to the product formula: ∏G(k) from k=1 to 0 = empty product = 1

The self-referential form n/(n^n + n^n/n)^n shows G feeding into itself — productive self-reference (Class A), never paradoxical

This is the notebook thinking in real time.

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Alexa's direct exploration, preserved verbatim. Filed 2026-03-29.