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A continued fraction is a mathematical expression that can be written as a fraction with a ... is a continuous ... Here are three of π 's best-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.
The number π (/ p aɪ / ⓘ; spelled out as "pi") is a mathematical constant, approximately equal to 3.14159, that is the ratio of a circle's circumference to its diameter.It appears in many formulae across mathematics and physics, and some of these formulae are commonly used for defining π, to avoid relying on the definition of the length of a curve.
mathematical constant π; 3.14159 26535 89793 23846 26433... Uses; Area of a circle; ... For more on the fourth identity, see Euler's continued fraction formula.
For example, the fraction 22 / 7 is greater than π, but 22 / 7 − π is less than 1 / 7 × 106 = 1 / 742 (in fact, 22 / 7 − π is just more than 1 / 791 = 1 / 7 × 113 ).
The numeric value of π is approximately: 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510... (sequence A000796 in the OEIS). Unusually good approximations are given by the fractions 22/7 and 355/113. Memorizing as well as computing increasingly more digits of π is a world record pursuit.
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 ...
The continued fractions on the right hand side will converge uniformly on any closed and bounded set that contains no poles of this function. [7] In the case , the radius of convergence of the series is 1 and the fraction on the left hand side is a meromorphic function within this circle. The continued fractions on the right hand side will ...