From: MatthewB2 on 18 Dec 2009 06:41 Hi, Is there a specific name for these summations besides "sums of radicals"? Is there a general formula for the value of the summation (for n less than infinity)? If there is no general formula, is there a website with the value of these summations calculated to a certain decimal precision? Do these summations all diverge?
From: Robert Israel on 18 Dec 2009 17:11 MatthewB2 <amintor1975(a)yahoo.com> writes: > Hi, > > Is there a specific name for these summations besides "sums of radicals"? > > Is there a general formula for the value of the summation (for n less than > infinity)? > > If there is no general formula, is there a website with the value of these > summations calculated to a certain decimal precision? > > Do these summations all diverge? x^(1/n) = exp(ln(x)/n) = 1 + ln(x)/n + O(1/n^2) so the sum diverges except in the trivial case x = 1. -- Robert Israel israel(a)math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
From: Raymond Manzoni on 19 Dec 2009 04:24 MatthewB2 a écrit : > Hi, > > Is there a specific name for these summations besides "sums of radicals"? > > Is there a general formula for the value of the summation (for n less than infinity)? > > If there is no general formula, is there a website with the value of these summations calculated to a certain decimal precision? > > Do these summations all diverge? Hi, Using the expansion proposed by Robert Israel and summing we may get following asymptotic expansion (for x>0) : S_n(x) ~ H_n^(1)*ln(x)/1! + H_n^(2)*ln(x)^2/2! + ... where H_n^(m) = sum_{k=1}^n 1/k^m Since H_n^(1) ~ ln(n) + gamma + O(1/n) and H_n^(m) ~ zeta(m) + O(1/n) for m integer > 1 (gamma is the Euler constant : <http://en.wikipedia.org/wiki/Euler–Mascheroni_constant> a better expansion of H_n is provided there) your asymptotic expansion may be rewritten : S_n(x) ~ ln(n)*ln(x) + F(ln(x)) + O(ln(x)^2/n) with F the function defined by : F(z)= gamma*z + sum_{k=2}^oo zeta(k)*z^k/k! (this implies that with the choice x^(1/n)-1-ln(x)/n instead of x^(1/n)-1 your sum would be convergent!) We could go further and write : S_n(x) ~ ln(n)*ln(x) + F(ln(x)) - ln(x)*(ln(x)-1)/(2*n) - ln(x)*(ln(x)^2-3*ln(x)+1)/(12*n^2) + O(1/n^3) and so on... I can't provide a name for this F function now but it is rather regular! Hoping it helped, Raymond
From: Raymond Manzoni on 19 Dec 2009 11:58 Raymond Manzoni a écrit : > F the function defined by : > F(z)= gamma*z + sum_{k=2}^oo zeta(k)*z^k/k! > > I can't provide a name for this F function now but it is rather regular! I'll just add that the Laplace transform of the F function is LF(s)= gamma/s^2 + sum_{n=2}^oo zeta(n)/s^(n+1) LF(s)= -psi(1-1/s)/s^2 = -(ln(Gamma(1-1/s)))' with psi the digamma function : <http://en.wikipedia.org/wiki/Digamma_function#Taylor_series> Fine continuation! Raymond
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