Series Everywhere

Calculus Level 4

Define N\large N as

N=6(1+11+3+11+3+5+11+3+5+7+...)21+124+134+144...\large N = \huge{ \frac{6\left(1+\frac{1}{1+3}+\frac{1}{1+3+5}+\frac{1}{1+3+5+7}+...\right)^2}{1+\frac{1}{2^4}+\frac{1}{3^4}+\frac{1}{4^4}...} }.

If there exists a positive number EE such that

E=N2+N2+N2+N2+N2+...\large E = \huge { \frac{N}{2+\frac{N}{2+\frac{N}{2+\frac{N}{2+\frac{N}{2+...}}}}}} ,

Find E2E^{2}.

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