(3)Note that letters other than are sometimesalso used; for example, the documentation for f,g, i, imin, imax in the uses.provide another method of expanding functions, namely as a ratio of two power series. The allows interconversion of continued fraction, power series, and rationalfunction approximations.A small sample of closed-form is given in the following table (cf.

In lieu of the above fraction. We may also call them regular continued fractions. Truncating the sequence at a.

The provide another fascinating class of continued fraction constants,and the is an example of a convergent generalized continued fraction functionwhere a simple definition leads to quite intricate structure.continued fractionvalueapproximateOEIS0.697774.0.581976.1.525135.1.541494.The value. Wolfram Web ResourcesThe #1 tool for creating Demonstrations and anything technical.Explore anything with the first computational knowledge engine.Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.Join the initiative for modernizing math education.Solve integrals with Wolfram Alpha.Walk through homework problems step-by-step from beginning to end. Zula europe download for free. Hints help you try the next step on your own.Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more.Knowledge-based programming for everyone.

.IntroductionContinued fractions are of great importance in many aspects, as they have many implementations for real problems where you want to describe something with an approximate fraction, or you simply want to replace a decimal or double number with a fraction.BackgroundAs always in mathematics the history of continued fractions are quite complicated. The earliest traces of continued fractions appear as far back as 306 b.c.

Other records have been found that show that the Indian mathematician Aryabhata used a continued fraction to solve a linear equation.In the western hemisphere it did not appear until the 17 th century, when William Brouncker and John Wallis started to formulate some form of continued fractions. The Dutch mathematician and astronomer, Christiaan Huygens made the first practical application of the theory in 1687, and wrote a paper explaining how to use convergents to achive the best rational approximations for gear ratios. These approximations enabled him to pick the gears with the best numbers of teeth as he was going to build a mechanical planetarium. Representation and calculations of continued fractionsHow to write a continued fraction anyway, well let's take the simple quadratic equation:x^2 - bx -1 = 0Move things over and divide by X and you can rewrite it like this:x = b + 1/xWe realize that we have a formula for X on the left, and we have an X on the right side. We now substitute the X on the right side with the formula on the right.

Sounds complicated, but here is the result with one substitution:x = b + 1/(b + 1/x)We can do this an infinite number of times and get a continued fraction. This is an interesting formula if we plug in b = 1 we would get the golden ration. (to get this number you simply type in (sqrt(5)-1)/2 in the program submitted).In the general representation of a continued fraction is given below, where x is a number (irrational or rational) and the coefficients a0, a1, and so on are all positive integers.The continued fraction is quite complicated to write in this way, so they are usually represented as a series of the coefficients as given below:Finite numbers would have a finite set of fractions, while irrational number (sqrt(2)) could be given a rational approximate representation if you adruptly stop it at one of the coefficients. Woutercx 21-Jul-Jul-12 11:01I was trying to solve problem 65 on projecteuler.net, your tool is wonderful, I just had to press buttons to find the answer. Except the answer was wrong.

I tried to put the number e, number of fractions = 99 (off by one, means 100). The 100th convergent was: 3647869 / 974616.Except3647869wasn't right. I know.NET is not very exact in mathematics.

Maybe the E number that.NET knows is not exact enough? - Update - No it isn't, but now I have a good starting point to solve the problem further.