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Since e is an irrational number (see proof that e is irrational), it cannot be represented as the quotient of two integers, but it can be represented as a continued fraction. Using calculus, e may also be represented as an infinite series, infinite product, or other types of limit of a sequence.
A continued fraction is an expression of the form = + + + + + where the a n (n > 0) are the partial numerators, the b n are the partial denominators, and the leading term b 0 is called the integer part of the continued fraction.
Euler derived the formula as connecting a finite sum of products with a finite continued fraction. (+ (+ (+))) = + + + + = + + + +The identity is easily established by induction on n, and is therefore applicable in the limit: if the expression on the left is extended to represent a convergent infinite series, the expression on the right can also be extended to represent a convergent infinite ...
Because the continued fraction expansion for φ doesn't use any integers greater than 1, φ is one of the most "difficult" real numbers to approximate with rational numbers. Hurwitz's theorem [22] states that any irrational number k can be approximated by infinitely many rational m / n with
The following list includes the continued fractions of some constants and is sorted by their representations. Continued fractions with more than 20 known terms have been truncated, with an ellipsis to show that they continue. Rational numbers have two continued fractions; the version in this list is the shorter one.
cofr - The sequence represents a continued fraction, for example the continued fraction expansion of e or π . cons - The sequence is a decimal expansion of a mathematical constant , such as e ( A001113 ) or π ( A000796 ).
For the continued fraction expansion. of a number, see simple continued fraction, of a function, see continued fraction This page was last edited on 11 ...
Using the explicit continued fraction expansion of e, one can show that e is not a Liouville number (although the partial quotients in its continued fraction expansion are unbounded). Kurt Mahler showed in 1953 that π is also not a Liouville number.