How do you/is it possible to express a=12+34+56+⋯ in the form pq(k+√n)? I'm still in high school, so I'm not familiar with especially sophisticated approaches for evaluating infinitely continued fractions - I've been able to set up a recurrence relation for everything I've encountered so far, but after writing it out as a(x)=xx+1+x+2x+3+⋯, I'm fairly sure that approach isn't going to work. I'm also not sure how to represent this in continued fraction notation, as each nested fraction has a unique numerator. It clearly converges to something, so can anyone point me in the right direction? :)
Answer
In view of the fact that the standard continued fraction expansion of e is
e=2+11+12+11+11+14+11+11+16+⋯,
I think you are mistaken in thinking that your number might even be algebraic, much less quadratic.
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