[Edited to fix typo]
Is there a precise formulation for when the sum
∞∑i=21i⋅f(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)=2√log(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=ℓ1i⋅f(i) and set ℓ to be the smallest positive integer so that f(ℓ)>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
f∘0(x)=xandf∘k+1(x)=f∘f∘k(x)
Note that if
fn(x)=n∏k=1log∘k(x)
then, for n>0,
∫∞exp∘n(1)dxxfn(x)log∘n(x)ainduction=∫∞exp∘n(1)dlog(x)fn(x)log∘n(x)ax↦exinduction=∫∞exp∘n−1(1)dxxfn−1(x)log∘n−1(x)ainduction=∫∞1dxxa+1induction=1a
Therefore, for all n≥0,
∫∞exp∘n(1)dxxfn(x)
diverges, yet for any a>0,
∫∞exp∘n(1)dxxfn(x)log∘n(x)a
converges. As far as logs and powers go, these border convergence/divergence pretty closely.
No comments:
Post a Comment