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This week we investigate a very interesting question about continued fractions. We start with the recursionf(n) = 1 / (1 + n*f(n+1)) for n > 1with the initial condition f(1) = a.Regardless of a and n, this leads tof(2) = 1/(1+2/(1+3/(1+4/... 1/(1+n*f(n+1))...)))for any n > 1.Note that f(n+1) can be iteratively computed using the recursion f(n+1) = (1/f(n) - 1)/nfor n > 0, with f(1) = a.The ParadoxIn the second formula, the one that expands f(2), let n tends to infinity. Will the right hand side of the formula converge? It looks like it won't, if you use the recursion (third formula) to compute f(n+1). It won't unless f(2) = 0.525135276...It seems to mean that unless f(2) = 0.525135276... then f(n) tends to -1/n as n tends to infinity, and there's no convergence. Also note that f(n) depends on f(1) = a, while the limit, for the continued fraction, does not. That's part of the paradox. The formula for f(2) is correct for any value of n and a, except n = infinity.Yet, if you assign a specific value to f(n), for any large value of n, and iteratively compute f(2) backwards using the first formula, you end up with f(2) being very, very close to 0.525135276...We did a test, computing f(n) forward starting with f(2) = -0.5, and ended up with f(353)=-0.0547... Doing it the other way around, starting with f(353) = 6 doing backward computations, we ended up with f(2) = 0.525132576... Isn't it fascinating?It turns out that the only value that guarantees convergence isf(2) = sqrt(pi*e/2)*erfc(1/sqrt(2))-1 = 0.525135276...This spreadsheet might help you understand the challenge.The ChallengeHow do you explain this paradox?Is the limiting value 0.525135276... a number that can be expressed using a finite number of elementary transcendental functions (trigonometric, log, exp, power, polynomials, hyperbolic and their inverses), rational numbers, and/or irrational numbers such as algebraic numbers and Pi, e, or the Euler constant? Most continued fractions such as the one in this challenge have a positive answer to this question: check out the Wolfram encyclopedia or Wikipedia for details. The AwardIf you are the first to successfully answer positively the second question in the Challenge section (that is, proving that such an expression exists) or negatively (such an expression does not exist, or you can prove that neither the result nor its opposite is provable), then you will earn $500,000 in real estate from me (basically, my house in the Seattle area will belong to you).Award for winning this challengeClick here to check out our previous challenges.DSC ResourcesCareer: Training | Books | Cheat Sheet | Apprenticeship | Certification | Salary Surveys | JobsKnowledge: Research | Competitions | Webinars | Our Book | Members Only | Search DSCBuzz: Business News | Announcements | Events | RSS FeedsMisc: Top Links | Code Snippets | External Resources | Best Blogs | Subscribe | For BloggersAdditional ReadingData Scientist Reveals his Growth Hacking Techniques10 Modern Statistical Concepts Discovered by Data ScientistsTop data science keywords on DSC4 easy steps to becoming a data scientist13 New Trends in Big Data and Data Science22 tips for better data scienceData Science Compared to 16 Analytic DisciplinesHow to detect spurious correlations, and how to find the real ones17 short tutorials all data scientists should read (and practice)10 types of data scientists66 job interview questions for data scientistsHigh versus low-level data scienceFollow us on Twitter: @DataScienceCtrl | @AnalyticBridgeSee More