5
votes
1answer
64 views

Number made from the first digits of $2^n$

Consider the number c made from the first digits of $2^n$. To be more precise, the n-th decimal digit of c is the first digit of $2^n$. The first digits from c are : ...
1
vote
3answers
281 views

Can every transcendental number be expressed as an infinite continued fraction?

Every infinite continued fraction is irrational. But can every number, in particular those that are not the root of a polynomial with rational coefficients, be expressed as a continued fraction?
11
votes
3answers
436 views

Closed form for a pair of continued fractions

What is $1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cdots}}}$ ? What is $1+\cfrac{2}{1+\cfrac{3}{1+\cdots}}$ ? It does bear some resemblance to the continued fraction for $e$, which is ...