Intellectual battle with limits begins!

Calculus Level 2

Ln=limx((x+a1)(x+a2)(x+an))1nx,L_n=\lim_{x\to\infty}((x+a_1)(x+a_2)\cdots(x+a_n))^{\frac{1}{n}}-x,

where ai=12i \large a_i = \frac{1}{2^i} .

Find limn(nLn)\displaystyle{\lim_{n\to\infty} (nL_n)}.

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