Hi, I need some help with some sequences. I need to solve a convergence problem, i.e. to find the conditions/bounds for an infinite sequence $\displaystyle \{ a_i \}_{i=1}^\infty$ of numbers $\displaystyle a_i \in \mathbb{R}^+$, such that the following two conditions hold:

$\displaystyle (a) \lim_{N \to \infty} \frac{1}{N^2} \sum_{i=1}^N {a_i} = 0$

$\displaystyle (b) \lim_{N \to \infty} \frac{1}{N} \sum_{i=1}^N \frac{1}{a_i} = 0.$

I already know one particular family of sequences, which is $\displaystyle \{ i^\alpha \}_{i=1}^\infty$ for $\displaystyle \alpha \in (0;1)$. I can show that for this family (a) and (b) hold, by bounding the sums by the integrals of their corresponding continuous functions, which I can then compute analytically.

My ultimate goal is to derive tight bounds for the sequence. I feel that my family of solutions already hints the bounds, which I think are $\displaystyle \alpha = 0$ and $\displaystyle \alpha = 1$ as a lower and upper bound. However, I'm not sure.

For example, I feel that the lower bound is given by (b), which should be that $\displaystyle a_i$ aproaches to infinity, however slowly, as i approaches infinity. Then, the upper bound can be derived from (a) by observing that a linear increase of $\displaystyle a_i$ makes the limit to be a constant, whereas anything sublinear ensures convergence to zero. I'd like a proof that is mathematically sound, however.

Any ideas will be greatly appreciated!!

EDIT: I, very sorry, but despite being careful in which forum to put the question, something must have gone wrong, and the post ended up in this forum. Please, can any moderator move it to "Number theory"? Or should I just delete it from here and repost it there?