Is it possible to find the limit of the sequence defined by: $$a_n = \left(1+ \frac{1}{2}\right)\left(1+\frac{1}{4}\right)\cdots\left(1+\frac{1}{2^n}\right)$$ I have proved that it converges.
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WolframAlpha comes up with this solution: \begin{align} \prod_{n=1}^\infty \left( 1 + \frac{1}{2^n} \right) &= \frac{(-1;\frac{1}{2})_\infty}{2} \\ &= 2.3842310290313\ldots \end{align} where $(a;q)_n$ is the $q$-Pochhammer symbol (link): $$ (a;q)_n = \prod_{k=0}^{n-1} (1 - a q^k) $$
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1Thanks. How can I prove that the limit is not $e$? – chen h. Oct 08 '14 at 13:14
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2You have $\ln(1+\frac{1}{2^n})<\frac{1}{2^n}$ for $n\geq 1$, hence if $P$ is your product you have $\ln(P)<\sum_{n\geq 1}\frac{1}{2^n}=1$. – Kelenner Oct 08 '14 at 13:35