My Math Forum Infinite series : 1/r^r

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 April 21st, 2010, 10:50 PM #1 Newbie   Joined: Apr 2010 Posts: 16 Thanks: 0 Infinite series : 1/r^r It is known that sigma(1/r^r) r=1,2,3.... = integral(1/x^x) limits 0 to 1.This is popularly known as the "Sophomore's dreams" Now the value of this sum turns out to be 1.2912859... While searching for some relation between this sum and pi i stumbled upon this : 1 + 1/(pis+sqrt(pis^2+1))=1.29129 where pis=pi/2. The two values differ only in the fifth decimal . Could there be any possible link between the two ???
 April 22nd, 2010, 12:32 AM #2 Math Team   Joined: Apr 2010 Posts: 2,780 Thanks: 361 Re: Infinite series : 1/r^r Hello Zeefinity, $\frac{1}{x^x}$ may be written as $\frac{1}{e^{x\ln{x}}$=$e^{-x\ln{x}$ I don't quite know how to integrate this function, so if some-one could give us a hand on that one, we can see whether Zeefinity's statement is right. Hoempa
 April 22nd, 2010, 05:13 AM #3 Newbie   Joined: Apr 2010 Posts: 16 Thanks: 0 Re: Infinite series : 1/r^r I searched the net regarding that integral and found out that it is a non integrable function whose integral cannot be expressed in terms of elementary functions.
 April 22nd, 2010, 05:25 AM #4 Math Team   Joined: Apr 2010 Posts: 2,780 Thanks: 361 Re: Infinite series : 1/r^r I tried in in Geogebra as well. It gave a questionmark when I tried it. Too bad. Hoempa
 August 28th, 2011, 11:18 PM #5 Senior Member   Joined: Aug 2011 Posts: 334 Thanks: 8 Re: Infinite series : 1/r^r Hello ! This is a link to a paper dealing with the integral of x^x : http://www.scribd.com/people/documents/ ... jjacquelin Then select "Sophomores Dream Function".

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