Ad
related to: continued fraction expansion of eeducation.com has been visited by 100K+ users in the past month
- Digital Games
Turn study time into an adventure
with fun challenges & characters.
- Education.com Blog
See what's new on Education.com,
explore classroom ideas, & more.
- Worksheet Generator
Use our worksheet generator to make
your own personalized puzzles.
- 20,000+ Worksheets
Browse by grade or topic to find
the perfect printable worksheet.
- Digital Games
Search results
Results from the WOW.Com Content Network
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 ...
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 ...
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.
His proofs are similar to Fourier's proof of the irrationality of e. In 1891, Hurwitz explained how it is possible to prove along the same line of ideas that e is not a root of a third-degree polynomial with rational coefficients, which implies that e 3 is irrational. [12] More generally, e q is irrational for any non-zero rational q. [13]
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.
The number e is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function.It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted .
A continued fraction for e x can be obtained via an identity of Euler: = + ... In this expansion, the rearrangement of the terms into real and imaginary parts is ...
Ad
related to: continued fraction expansion of eeducation.com has been visited by 100K+ users in the past month