<s>[INST] $(1-\frac{1}{2})(1+\frac{1}{3})(1-\frac{1}{4})(1+\frac{1}{5})...$ Is there any well-known value (Or Approximation) for this?
$$(1-\frac{1}{2})(1+\frac{1}{3})(1-\frac{1}{4})(1+\frac{1}{5})...$$
we know that it converges as $$\sum_{i=2}^{\infty}\frac{(-1)^{i+1}}{i}=ln2-1$$
So there is a trivial upper bound $\frac{2}{e}$ for it. Is there any better result? In addition is there any similar result for
$$(1-\frac{1}{2})(1-\frac{1}{4})(1-\frac{1}{8})(1-\frac{1}{16})...$$
or
$$(1+\frac{1}{2})(1+\frac{1}{4})(1+\frac{1}{8})(1+\frac{1}{16})...$$
 [/INST] HINT:
For the first question,
$$\left(1+\dfrac1{2n+1}\right)\left(1-\dfrac1{2n+2}\right)=1$$
$$\prod_{r=1}^n\left(1-\dfrac1{2^r}\right)=\dfrac{\prod_{r=1}^n(2^r-1)}{2^{1+2+\cdots+n}}$$
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