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 sum(e - eta^^k): convergence or divergence? Gottfried Ultimate Fellow     Posts: 789 Threads: 121 Joined: Aug 2007 08/17/2010, 11:45 AM (This post was last modified: 08/17/2010, 11:52 AM by Gottfried.) In the newsgroup news://sci.math I got a nice and concise answer by Prof. Israel. At 16.08.2010 with the subject: "Series : divergent or convergent?" I gave t=e=exp(1), b=t^(1/t) and the notation for the residual-term r_k = t - b^^k as example. This is the answer: (I inserted the correction of a wrong sign) Code:r_{k+1} = t - b^(b^^k) = t - b^(t - r_k)         = t - t b^(-r_k)  (since t = b^t)         = t (r_k ln(b) + O(r_k^2))   By the ratio test, the series will converge if |t ln(b)| < 1.   Since b^t = t says t ln(b) = ln(t), this is equivalent to 1/e < t < e. Your case t=e is on the boundary of this, so we need another term. r_{k+1} = t (r_k ln(b) - r_k^2 ln(b)^2/2 + O(r_k^3))         = r_k - r_k^2/(2 e) + O(r_k^3)              This fits with r_k ~ 2e/k, which would indicate that the sum diverges. [second post] In fact, I believe we should have      r_k ~= 2e/(k + ln(k)/(3 - 1/k))  as k -> infty. Again, it diverges. -- Robert Israel     Department of Mathematics     University of British Columbia Vancouver, BC, Canada I think that solves the problem. @Henryk: Shall I still copy the problem into the TPID-section? (or the math-facts/does this still exist?) Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread sum(e - eta^^k): convergence or divergence? - by Gottfried - 07/20/2010, 10:30 AM RE: sum(e - eta^^k): convergence or divergence? - by tommy1729 - 07/20/2010, 09:24 PM RE: sum(e - eta^^k): convergence or divergence? - by bo198214 - 07/24/2010, 11:38 PM RE: sum(e - eta^^k): convergence or divergence? - by tommy1729 - 08/16/2010, 11:05 PM RE: sum(e - eta^^k): convergence or divergence? - by Gottfried - 08/17/2010, 11:45 AM RE: sum(e - eta^^k): convergence or divergence? - by tommy1729 - 08/17/2010, 10:33 PM RE: sum(e - eta^^k): convergence or divergence? - by tommy1729 - 08/17/2010, 11:05 PM

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