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L=\mathop {\lim }\limits\left(1+a\right)\left(1+a^2\right)\left(1+a^4\right)+...+\left(1+a^{2^{n}}\right)
=\mathop {\lim }\limits\dfrac{1}{1-a}(1-a)\left(1+a\right)\left(1+a^2\right)\left(1+a^4\right)+...+\left(1+a^{2^{n}}\right)
=\mathop {\lim }\limits\dfrac{1}{1-a}(1-a^2)\left(1+a^2\right)\left(1+a^4\right)+...+\left(1+a^{2^{n}}\right)
=\mathop {\lim }\limits\dfrac{1}{1-a}(1-a^4)\left(1+a^4\right)+...+\left(1+a^{2^{n}}\right)
=...
=\mathop {\lim }\limits\dfrac{1}{1-a}\left(1-a^{2^{n}}\right)\left(1+a^{2^{n}}\right)
=\mathop {\lim }\limits\dfrac{1}{1-a}\left(1-a^{2^{n+1}}\right)
Do |a|<1\Rightarrow \mathop {\lim }\limits\left(1-a^{2^{n+1}}\right)=1-0=1
Vậy L=\dfrac{1}{1-a}.
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