[Math] Is $\{1,1,2,3,4,5,\cdots,i,\cdots \} $ the simple continued fraction algebraic or transcendental

Question

Is $$1+\cfrac{1}{1+\cfrac{1}{2+\cdots}} $$ or$\{1,1,2,3,4,5,\cdots,i,\cdots \} , i\in \mathbb{N}$ the simple continued fraction algebraic or transcendental?

Any reference is appreciated

EDIT and replacing $\{1,1,2,3,4,5,\cdots,i,\cdots \}$ with $\{1,2,2^2,2^3,2^4,2^5,\cdots,2^i,\cdots \} $ $i\in \mathbb{N}$ or any other patternful sequences that are upper unbounded ,can we get an algebraic number?

Answer 1

I believe this value is known to be transcendental; its value is known explicitly to be $1+\frac{I_1(2)}{I_0(2)}$, where $I_0()$ and $I_1()$ are the modified Bessel functions. For more details, see http://mathworld.wolfram.com/ContinuedFractionConstant.html . This continued fraction (and continued fractions with arithmetic progressions of coefficients in general) are related to certain Riccati equations, but unfortunately a bit of search on those terms doesn't really turn up any good lightweight accounts of the link between the two; you'll have to hunt about a bit more on your own for that.