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        riddles >> putnam exam (pure math) >> Limit Problem
        
(Message started by: THUDandBLUNDER on Jan 3^rd, 2007, 9:13pm)

Title: Limit Problem
Post by THUDandBLUNDER on Jan 3^rd, 2007, 9:13pm

Title: Re: Limit Problem
Post by towr on Jan 4^th, 2007, 2:30am

Taking an integration as approximation of the summation, I get
[hide] LN(sqrt(2) + 1) [/hide]
Not entirely sure whether they are the same in the limit though.

Title: Re: Limit Problem
Post by Icarus on Jan 4^th, 2007, 7:11pm

If we pull a factor of 1/n out of all terms, what we have is (1/n)(f(1/n) + f(2/n) + f(3/n) + ... + f(n/n)), where f(x) = 1/sqrt(1+x²). This is a Cauchy sum for f for the interval [0,1]. Hence it's limit is the integral of f(x) over that interval, provided the integral exists. Since f(x) = d/dx(arcsinh x), the limit is as towr gave.