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Lagrange's discovery implies that the canonical continued fraction expansion of the square root of every non-square integer is periodic and that, if the period is of length p > 1, it contains a palindromic string of length p − 1. In 1813 Gauss derived from complex-valued hypergeometric functions what is now called Gauss's continued fractions ...
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.
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 ...
Continued fraction for ζ(3/2) Jan 1, 2003: A080000: Number of permutations satisfying −k ≤ p(i) − i ≤ r and p(i) − i: Feb 10, 2003: A090000: Length of longest contiguous block of 1s in binary expansion of nth prime. Nov 20, 2003: A091345: Exponential convolution of A069321(n) with itself, where we set A069321(0) = 0. Jan 1, 2004: A100000
Since this continued fraction is infinite and every rational number has a terminating continued fraction, e is irrational. A short proof of the previous equality is known. A short proof of the previous equality is known.
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.
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.
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