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2 Corrected decimal approximation of number.

I recently discovered a constant that is constructed as follows:

$\chi=\sum_{n=1}^\infty (\frac{\cos{n}}{2|\cos{n}|}+\frac{1}{2}) 2^{-n}$

Furthermore I can prove that it is an irrational number whose decimal approximation is .048747846528...555609809015...

I conjecture that it is also a transcendental number but have not been able to prove this.

Anyone else want to take a crack at it? I am not a professional mathematician so if this number is not novel, then I appologize in advance.

One of the reasons I want to know this is that if this prototype number is proven transcendental than that would imply that an infinite set of similar transcendental numbers exists on the open interval (0,1) that are given by

$\chi_f=\sum_{n=1}^\infty (\frac{\cos{f(n)}}{2|\cos{f(n)}|}+\frac{1}{2}) 2^{-n}$

where f(n) is any algebraic function of n.

1

# Is this seemingly novel irrational constant also transcendental?

I recently discovered a constant that is constructed as follows:

$\chi=\sum_{n=1}^\infty (\frac{\cos{n}}{2|\cos{n}|}+\frac{1}{2}) 2^{-n}$

Furthermore I can prove that it is an irrational number whose decimal approximation is .048747846528...

I conjecture that it is also a transcendental number but have not been able to prove this.

Anyone else want to take a crack at it? I am not a professional mathematician so if this number is not novel, then I appologize in advance.

One of the reasons I want to know this is that if this prototype number is proven transcendental than that would imply that an infinite set of similar transcendental numbers exists on the open interval (0,1) that are given by

$\chi_f=\sum_{n=1}^\infty (\frac{\cos{f(n)}}{2|\cos{f(n)}|}+\frac{1}{2}) 2^{-n}$

where f(n) is any algebraic function of n.