11
votes
6answers
391 views

Asymptotic behaviour of a multiple integral on the unit hypercube

A few days ago I found an interesting limit on the "problems blackboard" of my University: $$\lim_{n\to +\infty}\int_{(0,1)^n}\frac{\sum_{j=1}^n x_j^2}{\sum_{j=1}^n x_j}d\mu = 1.$$ The correct claim, ...
2
votes
1answer
83 views

How to asymptotically estimate a lower bound of this function?

The function is given as $$f(x)\geq \sum_{i=1}^{[x/2]}f(i)+1$$ The boundary condition is $f(0)=0$. What I can get is this function grows faster than any polynomial function, and grows slower than ...