Friday, August 11, 2017

sequences and series - Convergence of suminftyi=2frac1icdotf(i)



[Edited to fix typo]



Is there a precise formulation for when the sum




i=21if(i) converges, in terms of the function f? Assume that f is smooth and monotonically increasing.



If f(i)ic for any c>0 then we know it converges. If f(i) is a constant then we know it doesn't. We can try functions in between. For example setting f(i)=2log(i) makes the sum converge but setting f(i)=log(i) makes it diverge according to Wolfram Alpha



There are of course a lot of functions so it might be hard to write a full classification. How about if we only including elementary functions that, for example, use only powers and logs?



Update. Is something like the following conjecture true? Consider i=1if(i) and set to be the smallest positive integer so that f()>0. The sum converges if and only if there exists c>0 such that f(i)clog(i)loglog(i)logloglog(i) where the log is applied an (as yet) unknown but fixed number of times.


Answer




Since f is monotonically increasing, we can use the integral test.



Define repeated composition by
f0(x)=xandfk+1(x)=ffk(x)



Note that if
fn(x)=nk=1logk(x)
then, for n>0,
expn(1)dxxfn(x)logn(x)ainduction=expn(1)dlog(x)fn(x)logn(x)axexinduction=expn1(1)dxxfn1(x)logn1(x)ainduction=1dxxa+1induction=1a
Therefore, for all n0,
expn(1)dxxfn(x)
diverges, yet for any a>0,
expn(1)dxxfn(x)logn(x)a
converges. As far as logs and powers go, these border convergence/divergence pretty closely.



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